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Evaluation and optimization of the quality of serviceperceived by mobile users for new services in cellular

networksMiodrag Jovanovic

To cite this version:Miodrag Jovanovic. Evaluation and optimization of the quality of service perceived by mobile users fornew services in cellular networks. Networking and Internet Architecture [cs.NI]. Télécom ParisTech,2015. English. NNT : 2015ENST0052. tel-01238450

https://hal.archives-ouvertes.fr/tel-01238450

https://hal.archives-ouvertes.fr

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H

È

S

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2015-ENST-0052

EDITE - ED 130

Doctorat ParisTech

T H E S E

pour obtenir le grade de docteur delivre par

TELECOM ParisTech

Specialite “Informatique et Reseau”

presentee et soutenue publiquement par

Miodrag JOVANOVIC

le 11 Septembre 2015

Titre

Evaluation et optimisation de la qualite de service percue parles utilisateurs mobiles pour les nouveaux services dans les

reseaux cellulaires

Directeur de these: Bartek BLASZCZYSZYNCo-encadrement de la these: Mohamed KARRAY

JuryM. Laurent DECREUSEFOND, Professeur, Telecom ParisTech President du juryM. Sem BORST, Professeur, Technische Universiteit Eindhoven et Bell Labs RapporteurM. Martin HAENGGI, Professeur, University of Notre Dame RapporteurM. Merouane DEBBAH, Professeur, Supelec et Huawei ExaminateurM. James ROBERTS, Professeur, IRT-SystemX ExaminateurM. Bartek BLASZCZYSZYN, Professeur, Ecole normale superieure et INRIA Directeur de these

TELECOM ParisTechecole de l’Institut Mines-Telecom - membre de ParisTech

46 rue Barrault 75013 Paris - (+33) 1 45 81 77 77 - www.telecom-paristech.fr

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To my family and friends

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RESUME: L’objectif de cette these est de developper des outils et des methodes pour l’evaluationde la qualite de service (Quality of Service - QoS) percue par les utilisateurs, en fonction de la demandede trafic, dans les reseaux cellulaire sans fil moderne. Ce probleme complexe, directement liee audimensionnement du reseau, implique la modelisation des processus dynamiques a plusieurs echelles detemps, qui en raison de leurs nature aleatoire se pretent a la formalisation probabiliste.

Tout d’abord, sur la base de la theorie de l’information, nous capturons les performances d’un seullien entre une station de base et un utilisateur dans un reseau cellulaire avec des canaux orthogonaux etla technologie MIMO. Nous prouvons et utilisons certaines bornes inferieures de la capacite ergodique envue de la theorie de l’information d’un tel lien, qui prend aussi en compte la variabilite du canal rapidecausee par la propagation des trajets multiples. Ces bornes donnent une base solide pour l’evaluationplus profonde de la qualite de service percue par les utilisateurs.

Ensuite, on considere plusieurs utilisateurs (eventuellement mobiles), arrivant dans le reseau et de-mandant un service. Nous considerons des services (elastiques) a debit variable dans lesquels les trans-missions de certaines quantites de donnees sont realisees d’une maniere ”best-effort”, ou services a debitconstant, dans lesquels une certaine vitesse de transmission doit etre maintenue pendant les periodesdemandees. Sur la base de la theorie des files d’attente, on capture cette demande du trafic et processusde service a l’aide des modeles appropries (multi-classes) de partage du processeur (processor-sharingPS) ou modele de perte. Dans cette these, nous adaptons les modeles PS existants et developpons unnouveau modele de perte pour le trafic streaming de transmission sans fil, ou les bornes theoriques (auregard de la theorie de l’information) mentionnees ci-dessus de la capacite des liens simples decriventles taux de service instantanes des utilisateurs. Les modeles multi-classes sont utilises pour capturerl’heterogeneite spatiale des canaux utilisateur. Ceux-ci dependent de l’emplacement geographique del’utilisateur et du phenomene de ”shadowing” de propagation.

Enfin, au-dessus des processus de file d’attente theoriques, il faut tenir compte d’un reseau multicel-lulaire, dont les stations de base ne sont pas necessairement regulierement placees, et dont la geometrieest en outre perturbee par la phenomene de shadowing. Nous abordons cet aspect aleatoire en utilisantdes modeles de geometrie stochastique, notamment processus de Poisson ponctuels et le formalisme dePalm applique a la cellule typique du reseau. En appliquant l’approche triple mentionnee ci-dessus,censee a representer tous les mecanismes cruciaux et les parametres de l’ingenierie des reseaux cellu-laires (tels que LTE - Long Term Evolution), nous etablissons des relations macroscopiques entre lademande de trafic et les metriques de la qualite de service percue par les utilisateurs pour certains ser-vices a debit binaire elastiques et constants. Ces relations sont obtenues principalement d’une manieresemi-analytique, c’est-a-dire qu’elles concernent des simulations statiques d’un processus ponctuel dePoisson (modelisation des emplacements des stations de base). Ceci afin d’evaluer ses caracteristiquesqui ne se pretent pas aux expressions analytiques.

Plus precisement, en ce qui concerne le trafic de donnees (le service de debit binaire elastique), nouscapturons l’interference inter-cellule, rendant les modeles des files d’attente PS de cellules individuellesdependantes, via un systeme d’equations de charge des cellules. Ces equations permettent de determinerle debit moyen par utilisateur, le nombre moyen d’utilisateurs et la charge moyenne de la cellule dansun grand reseau, en fonction de la demande du trafic. La distribution spatiale de ces metriques de QoSdans le reseau est egalement etudiee. Nous validons notre approche en comparant les resultats obtenusavec ceux mesures a partir de traces du reseau reel. Nous observons une concordance remarquable entreles predictions du modele et les donnees statistiques recueillies dans plusieurs scenarios de deploiement.

En ce qui concerne les services de debit binaire constants, nous proposons un nouveau modele stochas-tique pour evaluer la frequence et le nombre d’interruptions lors de streaming en temps reel en fonctiondes conditions radio utilisateur. Nous l’utilisons pour etudier les metriques de la qualite de service enfonction des conditions radio utilisateur dans les reseaux LTE.

Tous les resultats etablis ici contribuent au developpement de methodes de dimensionnement dereseau et sont actuellement utilises dans les outils internes d’Orange pour les calculs de capacite dureseau.

MOTS-CLEFS: QoS; LTE; debit; charge; theorie des files d’attente; geometrie stochastique; mesures;3GPP

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ABSTRACT: The goal of this thesis is to develop tools and methods for the evaluation of theQoS (Quality of Service) perceived by users, as a function of the traffic demand, in modern wirelesscellular networks. This complex problem, directly related to network dimensioning, involves modelingdynamic processes at several time-scales, which due to their randomness are amenable to probabilisticformalization.

Firstly, on the ground of information theory, we capture the performance of a single link betweena base station and a user in the context of a cellular network with orthogonal channels and MIMOtechnology. We prove and use some lower bounds of the information-theoretic ergodic capacity of sucha link, which account also for the fast channel variability caused by multi-path propagation. Thesebounds give robust basis for further user QoS evaluation.

Next, one considers several (possibly mobile) users, arriving in the network and requesting someservice from it. We consider variable (elastic) bit-rate services, in which transmissions of some amountsof data are realized in a best-effort manner, or constant bit-rate services, in which a certain transmissionrate needs to be maintained during requested times. On the ground of queuing theory, one captures thistraffic demand and service process using appropriate (multi-class) processor sharing (PS) or loss models.In this thesis, we adapt existing PS models and develop a new loss model for wireless streaming traffic,in which the aforementioned information-theoretic capacities of single links describe the instantaneoususer service rates. The multi-class models are used to capture the spatial heterogeneity of user channels,which depends on the user geographic locations and propagation shadowing phenomenon.

Finally, on top of the queueing-theoretic processes, one needs to consider a multi-cellular network,whose base stations are not necessarily regularly placed, and whose geometry is further perturbed bythe shadowing phenomenon. We address this randomness aspect by using some models from stochasticgeometry, notably Poisson point processes and Palm formalism applied to the typical cell of the network.

Applying the above three-fold approach, supposed to represent all crucial mechanisms and engineer-ing parameters of cellular networks (such as LTE), we establish some macroscopic relations between thetraffic demand and the user QoS metrics for some elastic and constant bit-rate services. These relationsare mostly obtained in a semi-analytic way, i.e., they only involve static simulations of a Poisson pointprocess (modeling the locations of base stations) in order to evaluate its characteristics which are notamenable to analytic expressions.

More precisely, regarding the data traffic (the elastic bit-rate service), we capture the inter-cellinterference, making the PS queue models of individual cells dependent, via some system of cell-loadequations. These equations allow one to determine the mean user throughput, the mean number of usersand the mean cell load in a large network, as a function of the traffic demand. The spatial distribution ofthese QoS metrics in the network is also studied. We validate our approach by comparing the obtainedresults with those measured from live-network traces. We observe a remarkably good agreement betweenthe model predictions and the statistical data collected in several deployment scenarios.

Regarding constant bit-rate services, we propose a new stochastic model to evaluate the frequencyand the number of interruptions during real-time streaming calls in function of user radio conditions.We use it to study the quality of service metrics in function of user radio conditions in LTE networks.

All established results contribute to the development of network dimensioning methods and arecurrently used in Orange internal tools for network capacity calculations.

KEY-WORDS: QoS; LTE; throughput; load; queueing theory; stochastic geometry; measures; 3GPP

Contents

1 Introduction 131.1 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Thesis contribution and structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Link quality 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 OFDM Cellular network with MIMO . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Link capacity given fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 Ergodic capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 MMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 MMSE capacity given fading . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.2 MMSE ergodic capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 MMSE-SIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Numerical results for the link capacity . . . . . . . . . . . . . . . . . . . . . . . . 262.5.1 Link layer model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.2 Comparison to simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.3 Comparison to measurements . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.4 Approximate link quality estimation via simulations . . . . . . . . . . . . 28

2.6 Link quality observed by a typical user . . . . . . . . . . . . . . . . . . . . . . . . 302.6.1 SINR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6.2 Spectral efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Appendices 352.A Theoretical results: MIMO flat-fading channel with additive noise . . . . . . . . 35

2.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.2 Capacity lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.3 Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 User throughput versus traffic demand — global network performance via afixed point problem 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Related work regarding the dimensioning problem . . . . . . . . . . . . . 433.2.2 Related work regarding QoS evaluation . . . . . . . . . . . . . . . . . . . 44

3.3 Processor-sharing queue model for one cell scenario . . . . . . . . . . . . . . . . . 453.3.1 No mobility case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5

6 CONTENTS

3.3.2 Infinite mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4 Multi-cell scenario: symmetric networks . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.1 Cell load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Multi-cell irregular networks scenario . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.2 Generalization of processor sharing model . . . . . . . . . . . . . . . . . . 533.5.3 Global network characteristics . . . . . . . . . . . . . . . . . . . . . . . . 553.5.4 Mean cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Heterogeneous networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.6.2 Typical and mean cell in multi-tier network . . . . . . . . . . . . . . . . . 623.6.3 Full interference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.6.4 Weighted interference model . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.7 Numerical results: network dimensioning and QoS estimation . . . . . . . . . . . 693.7.1 Validation in a dynamic context . . . . . . . . . . . . . . . . . . . . . . . 703.7.2 Hexagonal LTE network dimensioning . . . . . . . . . . . . . . . . . . . . 723.7.3 Mean performance estimation of irregular network using Poisson process . 753.7.4 Numerical results for heterogeneous networks . . . . . . . . . . . . . . . . 813.7.5 Spatial distribution of QoS parameters averaged over many cells in the

network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Appendices 933.A Proof of Proposition 4 in the Markovian case . . . . . . . . . . . . . . . . . . . . 93

4 Quality of service in real-time streaming 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 Streaming in wireless cellular networks . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3.1 System assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3.3 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4 Quality of real-time streaming in LTE . . . . . . . . . . . . . . . . . . . . . . . . 1064.4.1 LTE model and traffic specification . . . . . . . . . . . . . . . . . . . . . . 1074.4.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Appendices 1214.A A general real-time streaming (RTS) model . . . . . . . . . . . . . . . . . . . . . 121

4.1.1 Traffic demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.1.2 Resource constraints and outage policy . . . . . . . . . . . . . . . . . . . 1224.1.3 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.1.4 Mathematical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Conclusion and future work 127

List of Figures

2.1 Jensen (2.8) and asymptotic (2.7) lower bounds for the capacity (peak bit-rate) ofthe MIMO flat fading channel with additive noise in the downlink of an OFDMAcellular network. Capacity as function of the distance between the user and hisserving base station. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Performance of SISO without fading evaluated using analytic approximation (2.15)and link simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Analytic relation of the peak bit-rate to SINR compared to 3GPP simulation . . 28

2.4 Analytic relations of the peak bit-rate to SINR compared to measurements; seeSection 2.5.3 for the details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Simulations versus the analytical expression (right-hand side of (2.19)) for thecalibration case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 CDF of the coupling gain (antenna gain minus propagation loss) . . . . . . . . . 31

2.7 CDF of SINR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 CDF of normalized user throughput . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9 Comparison of the analytic capacity and the asymptotic formula . . . . . . . . . 39

3.1 Load versus traffic demand per cell . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Mean user throughput versus traffic demand per cell . . . . . . . . . . . . . . . . 72

3.3 95% quantile of user throughput versus traffic demand . . . . . . . . . . . . . . . 72

3.4 Mean user throughput as function of the cell radius for different load situations . 73

3.5 Mean user throughput as function of the cell radius for different traffic demanddensities (adapted load) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 Cell radius versus traffic demand density for mean user throughput 104kbit/s . . 74

3.7 Cell radius versus traffic demand density for different mean user throughputs . . 75

3.8 Local user throughput versus local traffic demand for some zone (selected to sat-isfy a spatial homogeneity of the base stations) of an operational cellular networkdeployed in a big city in Europe. 9288 different points correspond to the mea-surements made by different sectors of different base stations during 24 differenthours of some given day. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.9 Cell load and the stable fraction of the network versus traffic demand per cell inthe full interference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.10 Number of users per cell versus traffic demand per cell in the full interference model. 77

3.11 Mean user throughput in the network versus traffic demand per cell in the fullinterference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.12 Load and the stable fraction of the network versus traffic demand in the weightedinterference model. Also, load estimated from real field measurements. . . . . . . 78

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8 LIST OF FIGURES

3.13 Number of users versus traffic demand per cell in the weighted interference model.Also, the same characteristic estimated from the real field measurements. . . . . 79

3.14 Mean user throughput in the network versus traffic demand per cell in the weightedinterference model. Also, the same characteristic estimated from the real fieldmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.15 Mean user throughput in the network versus traffic demand per area for an urbanzone of a big city in Europe. (The density of base stations is 4 times smaller thanin the dense urban zone considered in Figure 3.14). . . . . . . . . . . . . . . . . . 80

3.16 Ripley’s L-function calculated for the considered dense urban and urban networkzones. (L function is the square root of the sample-based estimator of the expectednumber of neigbours of the typical point within a given distance, normalized bythe mean number of points in the disk of the same radius. Slinvyak’s theoremallows to calculate the theoretical value of this function for a homogeneous Pois-son process, which is L(r) = r.) In fact, in large cities spatial, homogeneous“Poissonianity” of base-station locations is often satisfied “per zone” (city center,residential zone, suburbs, etc.). Moreover, log-normal shadowing further justifiesthe Poisson assumption, cf. [20, 29] . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.17 Accuracy of the homogeneous approximation of the mean cell . . . . . . . . . . . 823.18 Cell load versus traffic demand per cell in the full interference model. . . . . . . . 833.19 Number of users per cell versus traffic demand per cell in the full interference model. 833.20 Mean user throughput in the network versus traffic demand per cell in the full

interference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.21 Cell load versus traffic demand per cell in the weighted interference model. . . . 853.22 Number of users per cell versus traffic demand per cell in the weighted interference

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.23 Mean user throughput in the network versus traffic demand per cell in the weighted

interference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.24 CDF of BS powers in the operational network in the downtown of a big city (blue)

and normal distribution approximation (red). . . . . . . . . . . . . . . . . . . . . 883.25 CDF of cell load for the downtown of a big city obtained either from the vari-

able power model, from real-field measurements, or from the model where thetransmitted powers are assumed constant. . . . . . . . . . . . . . . . . . . . . . . 89

3.26 CDF of the mean users number for the downtown of a big city. . . . . . . . . . . 903.27 CDF of the throughput for the downtown of a big city. . . . . . . . . . . . . . . . 903.28 CDF of cell load for a mid-size city obtained either from the variable power model,

from real-field measurements, or from the model where the transmitted powers areassumed constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.29 CDF of the mean number of users for the mid-size city. . . . . . . . . . . . . . . 913.30 CDF of the throughput for the mid-size city. . . . . . . . . . . . . . . . . . . . . 91

4.1 Cumulative distribution function of the SINR . . . . . . . . . . . . . . . . . . . . 1084.2 Fraction of time in outage; traffic 900 Erlang/km2 . . . . . . . . . . . . . . . . . 1104.3 Fraction of time in outage; traffic 600 Erlang/km2 . . . . . . . . . . . . . . . . . 1114.4 Number of outage incidents; traffic 900 Erlang/km2 . . . . . . . . . . . . . . . . . 1124.5 Number of outage incidents; traffic 600 Erlang/km2 . . . . . . . . . . . . . . . . . 1134.6 Deep outage versus outage time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.7 Mean total throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.8 Deterministic vs Poisson arrivals; fraction of time in outage, 900 Erlang/km2 . . 1164.9 Deterministic vs Poisson arrivals; fraction of time in outage, 600 Erlang/km2 . . 117

LIST OF FIGURES 9

4.10 Deterministic vs Poisson arrivals; number of outage incidents, 900 Erlang/km2 . 1184.11 Deterministic vs Poisson arrivals; number of outage incidents, 600 Erlang/km2 . 119

10 LIST OF FIGURES

List of Tables

2.1 Results of the linear fittings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Cell spectral efficiency: Comparison of the 3GPP simulations and the analytic

results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Mean and standard deviation of spatial distribution of QoS metrics for the down-town of a big city . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2 Mean and standard deviation of spatial distribution of QoS metrics for the mid-sizecity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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12 LIST OF TABLES

Chapter 1

Introduction

1.1 Thesis motivation

There is a need for simple, yet realistic methods for the evaluation of the quality of service(QoS) in wireless networks capturing both the spatial distribution of the elements of the networkand the temporal dynamics of users and having a limited number of parameters. This can beobtained by decomposing the problem into three layers corresponding to different time-scales,which are addressed on the ground of information theory, queuing theory and stochastic geometry.Firstly, information theory studies the performance of a single radio link accounting particularlyfor the signal variations due to multi-path fading. Once the link performance is characterized,resources (power, bandwidth etc.) are allocated to the users while accounting for their mutualinterference. This can be modelled by an appropriate service policy on the ground of queuingtheory which accounts next for the users’ arrivals, mobility and departures and allows appropriatetime averages. Finally, stochastic geometry is used to model network, i.e. base stations spatialpattern and shadowing.

Individual elements of the above puzzle (i.e. information theory, queuing theory and stochas-tic geometry models) are often studied and optimized separately. The main specificity of themethodology proposed in this thesis is a global approach that combines these three elements. Indoing so, it is necessary to separate carefully the time scales of different elements of the networkdynamics.

We apply an information-theoretic modeling of a link layer between a user and a base station.In this context we show that the worst additive noise is the white Gaussian one and establish alower bound for the link capacity. Further the modeling approach consists in representing theconfiguration of users (positions, call durations or volumes, allocated resources) as a randomobject (point pattern with associated random variables), which evolves in time. The quality ofservice perceived by the users may then be expressed as a function of the stationary state of thisprocess and thus will depend only on its distribution parameters. This approach often allowsone for an explicit evaluation of the key QoS characteristics and for efficient optimization of thenetwork cost and capacity. Some examples from other research works which prove the pertinenceof this approach, can be found in [49], [78], [24], [68], [42], [10], [54]. We use homogeneousspatial Poisson point processes to model base station positions and apply results from stochasticgeometry to evaluate QoS.

The performance of wireless cellular networks is often evaluated in terms of parameters suchas the spectral efficiency [7] (in particular within 3GPP [3]) or the outage probability [44].However from the point of view of an operator, it is even more important to calculate the QoS

13

14 CHAPTER 1. INTRODUCTION

perceived by the users; and in particular to relate this QoS to the key network parameters suchas the traffic demand, the cell radius, the transmitted power, etc. This relation is crucial for thenetwork dimensioning; i.e., evaluating the minimal number of base stations (more generally, therequired network setup) assuring some QoS (for some given traffic demand). This permits inparticular to minimize the network cost. The probabilistic approach described above often allowsan explicit evaluation of the key characteristics such as users QoS and for efficient optimizationof the network dimensioning.

Classically, the services are classified into two main categories:

• Variable bit-rate (VBR); e.g., mail, ftp. Users aim to transmit some given volume of dataat a bit-rate which may be decided by the network

• Constant bit-rate (CBR); e.g., voice, video conferencing. Users require some given (con-stant) bit-rate for some duration. In this case the requested bit-rates may sometimes exceedthe available capacity, a situation usually called congestion. CBR services do not toleratetemporary interruptions of their transmissions. Consequently, if congestion occurs, the net-work blocks (i.e., refuses the access to) new calls and/or drops (i.e., interrupts definitely)some other calls during their transmissions.

When we account for calls’ arrivals, mobility and departures, the QoS perceived by theusers (in the long run of the network) is different for each of the above traffic classes. For VBRconnections, the QoS may be defined in terms of the mean throughput or delay per user [25], [59].For CBR calls, the main QoS indicators are the blocking and dropping probabilities [10], [42], [59].The research done in the last few years permitted to build efficient methods to calculate theseQoS indicators (see for example [6]). These methods are based respectively on processor sharingfor VBR and on multi-class Erlang models for CBR services.

However, new multimedia services are gaining interest in wireless cellular networks, especiallystreaming services [45]. Streaming connections require some given bit-rate for some duration [68],[75]. Thus congestion may occur (when the bit-rates requested by the users in the network exceedthe available capacity). All streaming calls are admitted, but, as a counterpart, they toleratetemporary interruptions of their transmission. We distinguish two sub-classes:

• Streaming-RT (Real-Time): e.g. mobile TV, RTP streaming. When congestion occurs, thecorresponding portions of some calls are definitely lost, but the call is not dropped.

• Streaming-NRT (Non-Real-Time): e.g., streaming-video (youtube, dailymotion, on de-mand video) on the web. When congestion occurs, the corresponding portions of calls aredelayed.

For the streaming users, the QoS is related to the frequency of the interruption of their callsand the durations of these interruptions. These performance measures depend strongly on themobility of users, as mobility increases the variability of the radio conditions.

1.2 Thesis contribution and structure

This thesis comprises three technical chapters. In Chapter 2 we focus on the first element ofthe analytic approach: single link quality between a user and a base station. We examineradio links in cellular networks such as LTE (Long Term Evolution) and HSDPA (High-SpeedDownlink Packet Access) and take into account MIMO (Multiple Input Multiple Output) andOFDM (Orthogonal Frequency Division Multiplex) technologies. The principal result is a lower

1.2. THESIS CONTRIBUTION AND STRUCTURE 15

theoretical bound for a single link quality which is very close to the exact link quality and whichis tractable analytically; i.e. calculated in a simple manner.

Once expressions for link quality are developed, we take into account user dynamics usingqueuing theory. Finally, we apply the results from stochastic geometry to model the spatialconfiguration of network resources and users. In Chapter 3 and Chapter 4 we apply the abovementioned methodology to evaluate QoS for the different types of services. Namely, in Chapter 3we develop a method for the user throughput estimation in large cellular networks, regular orirregular, for VBR traffic. In addition, we could estimate the mean number of users and the meancell load in the network. All the above means account for the disparity of different base stationsand traffic randomness over some period of time (one hour, for example). Further, we are ableto estimate the spatial CDF (Cumulative Distribution Function) of mean user throughput, meannumber of users and cell load over all base stations (averaged in time).

The core of the mathematical modeling in Chapter 3 was to capture the dependence betweenthe traffic demand and the interference in cellular networks with orthogonal channels (in timeand/or frequency). We did this using the aforementioned probabilistic tools in order to getanalytically tractable and simple relations, but in a manner that reflects the physical behaviourof the system. It turns out that the dependence between the traffic demand and the interferenceis well captured via a fixed-point problem. Solving this problem, we get all elements to evaluateQoS as function of traffic demand.

In Chapter 4 the evaluation of the QoS for real-time streaming is presented. The numberand duration of interruptions are calculated as function of traffic demand and radio conditions,i.e. SINR (Signal to Interference and Noise Ratio). Hexagonal cellular network with orthogonalchannels is considered. The stochastic analysis is based on Poisson processes representation ofthe traffic and Palm formalism related to the typical call. The results can be used to estimatethe QoS for this type of traffic, but also for network dimensioning. In fact, based on the spatialdistribution of radio conditions we can deduce the QoS at all positions in a cell (area servedby one base station) for a given traffic demand. On the other hand the spatial distribution ofSINR depends on cell radius. So, determining a constraint on QoS, one can deduce what is thenecessary cell radius to satisfy this constraint on QoS. This manipulation can be done for anyvalue of traffic demand, which is the cellular network dimensioning.

Chapter 2 is based on the following publications [61] and [63]. Chapter 3 is constructed fromthe following articles [62], [15], [57] and [16] and the Chapter 4 is an adapted version of [17].

The results of the thesis are used in Orange tools, such as the operational tool Utrandim,for the dimensioning of wireless cellular networks. They are also used to study the spectral andenergy efficiencies and the required emitted power in these networks.

16 CHAPTER 1. INTRODUCTION

Chapter 2

Link quality

2.1 Introduction

In this Chapter we are interested in the link quality analysis. A link is a communication channelbetween two or more communicating devices. We need it to develop a global analytic approachto the performance evaluation of wireless cellular networks and especially LTE networks.

The link performance in an OFDM (used in LTE) cellular network with MIMO antennas maybe studied using two methodologies. Information theory considers the ultimate performance ofbest possible coding schemes and looks for mathematical formulae to describe this performance.Real systems, such as 3GPP (3rd Generation Partnership Project) [3], deploy suboptimal codingschemes which are usually evaluated by simulation.

A key link characteristic of OFDM cellular networks is the peak bit-rate at each locationdefined as the maximal bit-rate a user can get at the considered location from his serving basestation. The objective of the present Chapter is to establish some closed form informationtheoretic bounds for the peak bit-rate in OFDM cellular networks with MIMO and compare themto real system performance predicted by simulation and estimated from field measurements.

We describe a simple model of a MIMO cellular network which permits to obtain an analyticalexpression of users’ bit-rates, which are feasible from the information theory point of view.This expression accounts for the variety of MIMO configurations (numbers of transmitting andreceiving antennas) and radio conditions (SINR). This expression is compared to practical LTEperformance evaluated by 3GPP simulations for different cases including the so-called calibrationcase [3]. The comparison shows that the analytical expression may be adjusted to the practicalperformance by a multiplicative coefficient, which depends on the MIMO configuration but noton the SINR. Additionally, we show the progress margin for potential evolution of the technology.

The capacity of the MIMO channel without interference is known. Accounting for the extra-cell interference, Proposition 2 gives a lower bound for the downlink capacity in a multi-cellOFDM network with MIMO antennas. This bound relies on the observation made in Proposi-tion 3 that the worst additive noise for the capacity of the MIMO flat-fading complex-valuedchannel is the white Gaussian one. In order to make the established lower bound more explicit,we give an asymptotic approximation based on random matrix theory and derive also a furtherlower bound from Jensen’s inequality.

Finally we build bounds for the peak bit-rate of the MMSE (Minimum Mean Square Error)scheme currently implemented in operational networks as well as its improvement MMSE-SIC(Successive Interference Cancellation).

17

18 CHAPTER 2. LINK QUALITY

2.2 Related work

Telatar [91] gives the information theoretic capacity of a MIMO channel with fading and AWGN(Additive White Gaussian Noise). Different MIMO configurations are compared for this channelby Foschini and Gans [46]. Blum et al. [22] study the capacity of a MIMO cellular network withflat Rayleigh fading. Clark et al. [32] show that in an OFDM system with a sufficiently largenumber of sub-carriers, the capacity with respect to Rayleigh fading is approximately normallydistributed. Tulino and Verdu [94] apply random matrix theory to analyze this capacity. Randommatrix theory is useful to study CDMA as for example in [93], [95] and [43].

The 3GPP [3] evaluates the performance of LTE systems by simulation. Goldsmith andChua [53] observed that real coding schemes performance may be described by a modificationof the famous log2 (1 + SNR) Shannon’s formula. Mogensen et al. [73] have observed that theLTE capacity in the AWGN context is well approximated by this formula with a multiplicativecoefficient. These ideas will be extended in the present Chapter to MIMO cellular networks withfading.

Explicit expressions for the capacity of AWGN channels are well known. Interference inwireless cellular networks is not necessarily Gaussian nor white (the term white means thatthe samples are independent and identically distributed). The explicit expression for capacityin such context is not known. To circumvent this difficulty, a possible idea is to look for alower bound and check whether it is tight enough to meet a desired precision. Using a resultof Shannon [84, Theorem 18], it may be shown that, in a SISO channel, the worst additivenoise process with given power is AWGN. This result is extended to a network with relays byShomorony and Avestimehr [85]. Diggavi and Cover [40] study the worst noise process for anadditive channel under covariance constraints and characterize the so-called saddle-point inputand noise distributions for the mutual information [40, Theorem II.1]. Girnyk et al. [51] calculatethe asymptotic sum-rate of uplink MIMO cellular network.

Note that the fact that the worst additive noise is Gaussian may be derived from [40, The-orem II.1]; but the input and the noise vectors are real-valued there whereas we shall considercomplex-valued random vectors. The whiteness of the worst noise process proved in Proposition 3does not follow immediately from the aforementioned result either.

2.3 OFDM Cellular network with MIMO

2.3.1 Network model

We consider a wireless network composed of several base stations (BS). The power transmitted byeach BS is limited to some given maximal value. The network operates the Orthogonal Frequency-Division Multiple Access (OFDMA) linked to OFDM, which we describe now. The frequencyspectrum allocated to the considered network is divided into a given number of sub-carriers,which are made available to all base stations. Each BS allocates disjoint subsets of the sub-carriers to its users. Each user is served by a single BS and receives only other-BS interference;that is the sum of powers emitted by other BS on the sub-carriers allocated to him by his servingBS. We consider multiple input and multiple output (MIMO) antennas. More precisely, BS areequipped with tA transmitting antennas whereas users have rA receiving antennas and each BSuses all its transmitting antennas to serve a given user.

We assume that the bandwidth of each sub-carrier is smaller than the coherence frequencyof the channel, so we can consider that the fading in each sub-carrier is flat [23]. That is, theoutput of the channel at a given time depends on the input only at the same instant of time.Indeed, the use of a cyclic prefix in OFDM permits to transform the frequency selective fading

2.3. OFDM CELLULAR NETWORK WITH MIMO 19

channel into a set of parallel flat fading channels [92, §3.4.4]. We don’t make any assumptionon the correlation of the fading processes corresponding to different subcarriers for a given userand a given BS. However, the fading processes for different users or base stations are assumedindependent.

Time is divided into time-slots of length smaller than the coherence time of the channel, sothat, for a given sub-carrier, the fading remains constant during each time-slot and the fadingprocess in different time-slots may be assumed ergodic. Such model for fading generalizes theso-called quasi-static model where the fading process at different time-slots is assumed to beindependent and identically distributed. We shall always assume that the receiver knows thefading.

The codeword duration equals the time-slot, which is assumed sufficiently large so that thecapacity (peak bit-rate) within each time-slot may be defined in the asymptotic sense of theinformation theory.

Users perform single user detection; thus the interference from other BS is added to AWGN.The statistical properties of the interference are not known a priori since they depend on thecoding of other BS.

Consider a user served by a BS indexed by u. For a given sub-carrier and time-slot, weconsider the following discrete-time model of the OFDM channel with MIMO [23, Equation (7)]

Yn = L−1/2u HuΨn + Θn + In, n ∈ N (2.1)

where n is a discrete-time index, Yn ∈ CrA is the channel output, Ψn ∈ CtA is the channel inputsignal, Hu is a complex matrix of dimension rA× tA representing the fading with the serving BSu, In ∈ CrA is the interference, Θ1,Θ2, . . . are i.i.d. random noises with values in CrA such thateach Θn is circularly-symmetric Gaussian with covariance matrix E[ΘnΘ∗n] = N IrA where Θ∗ndesignates the transpose complex conjugate of Θn, N is a given positive constant and IrA is theidentity matrix of dimension rA, and Lu is the propagation loss due to distance and shadowingbetween the user and BS u. The propagation loss Lu is the ratio between the emitted and

received powers, hence the factor L−1/2u in Equation (2.1).The interference equals to

In =∑v 6=u

L−1/2v HvΨv,n

where the sum is over the interfering BS v 6= u, Ψv,n is the transmitted signal by the interferingBS v, Hv represents fading for BS v, and Lv is the propagation loss due to distance and shadowingbetween the user and BS v.

We make the following probabilistic assumptions:

(H1) All the channel input signals are centred; i.e. E [Ψn] = 0.

(H2) The signals transmitted by different antennas including multiple antennas of the sameBS are independent. Let P be the power transmitted by each BS in a given sub-carrieraggregated over all the tA transmitters. Assume that this power is equally partitionedbetween the tA transmitting antennas; each one emitting a power P/tA. This assumptionis justified by the last statement in Proposition 3.

(H3) The fading matrices Hv are constant for all channel uses n ∈ N within a given time-slotand sub-carrier. For a given BS v, the fading matrix Hv is resampled across different time-slots, and we assume that it follows a stationary and ergodic sequence of random matrices.Moreover, these processes are independent across different BS v.

(H4) Users are motionless at the considered information theoretic time-scale; that is Lv areconstant for all BS v and time-slots.


2.3.2 Link capacity given fading

In this section, we focus on a given time-slot and sub-carrier; that is, the expectation E [·] is takenwith respect to the distribution of the transmitted signals and noise. By Assumptions (H1-H2),the covariance matrix of the transmitted signals are

E [ΨnΨ∗n] = E[Ψv,nΨ∗v,n

]=P

tAItA (2.2)

and the covariance matrix of the interference equals

E [InI∗n] = E

∑v 6=u

L−1v HvΨv,nΨ∗v,nH

∗v

=∑v 6=u

L−1v HvE

[Ψv,nΨ∗v,n

]H∗v =

P

tA

∑v 6=u

HvH∗v

Lv

Noise and interference are assumed independent, thus the covariance matrix of Σn = Θn+Inis

∆=E [ΣnΣ∗n] = N IrA +P

tA

∑v 6=u

HvH∗v

Lv(2.3)

The capacity of a channel may be interpreted as a maximal average bitrate sustainable in longcommunication time. The capacity C is defined in Section 2.6.2.

Proposition 1 The capacity C of the OFDM channel with MIMO (2.1) with power constraint (2.2)is lower bounded by

C ≥ log2 det

(IrA +

P

tA

HuH∗u

Lu∆−1

)(2.4)

where the noise plus interference covariance matrix ∆ is given by (2.3).

Proof. The mathematical background and proof are defended in Section 2.6.2.

We call the right-hand side of the above equation1 feasible bit-rate for the considered user.Since our assumptions (H1)-(H5) are the same for all users, we get similar expressions for thefeasible bit-rates of the other users and this collection of bit-rates of the different users is feasible.

Remark 1 Continuous-time. Consider a continuous-time model of the channel (2.1). Let wbe the bandwidth of the considered sub-carrier. The results in the discrete-time extend to thecontinuous-time case, but the capacity bounds, such as the right-hand side of (2.4), should bemultiplied by the bandwidth w of the considered sub-carrier.

2.3.3 Ergodic capacity

Consider now a given sub-carrier and multiple time-slots. Recall that we assumed that the fadingmatrices are ergodic across different time-slots. Then, by the ergodic theorem, the capacityaveraged over a large number of time-slots approaches the ergodic capacity E [C] where theexpectation is taken with respect to the fading distribution.

1which is consistent with [22, Equation (2)]

2.3. OFDM CELLULAR NETWORK WITH MIMO 21

Corollary 1 The ergodic capacity of the OFDM channel with MIMO (2.1) is lower bounded by

E [C] ≥ E[log2 det

(IrA +

P

tA

HuH∗u

Lu∆−1

)](2.5)

where ∆ is given by (2.3).

Proof. The result follows by taking the expectation of Equation (2.4) with respect to thefading.

Asymptotic bound

The right-hand side of the Equation (2.5) may be approximated using the following asymptoticresult when the number of transmitting and receiving antennas goes to infinity. As will beshown in Appendix 2.1.3, the approximation remains reasonable even for a moderate number ofantennas.

Lemma 1 [69, Appendix] Assume that the fading matrix of each base station has i.i.d. centredcomponents with variance 1. (Recall that we have already assumed that Hv are independentacross v.) Assume that tA, rA →∞ such that tA

rA→ Q, then

1

rAlog det

IrA +P

tA

HuH∗u

Lu

N IrA +P

tA

∑v 6=u

HvH∗v

Lv

−1 (2.6)

converges almost surely to

Q∑v 6=u

log

(Lv + P

Nη1

Q

Lv + PNη2

Q

)+Q log

(1 +

P

NLuη1

Q

)

+ log

(η2

η1

)+ η1 − η2 (2.7)

where η1 and η2 are respectively solutions of

η1 +∑v

Pη1

PQη1 +NLv

= 1

η2 +∑v 6=u

Pη2

PQη2 +NLv

= 1

The above expressions involve solutions of two non linear equations, which require the knowl-edge of the received powers from all interfering base stations. In what follows we will establishanother lower bound for the capacity, whose evaluation is much simpler, as simple as the evalu-ation of the capacity of the AWGN channel, and requires only the knowledge of the interferencepower aggregated over all the interfering base stations. We shall compare the two bounds nu-merically in Section 2.3.3 below.


Jensen’s bound

The following proposition gives a lower bound for the ergodic capacity under the assumptionthat the covariance of the fading matrix Hv equals identity; that is

E [HvH∗v ] = IrA , for all BS v

which means in particular that the fadings of two different transmitting antennas are uncorre-lated.

Proposition 2 Assume that E [HvH∗v ] = IrA , for all base station v, then the ergodic capacity

E [C] of the channel (2.1) is lower bounded by

E [C] ≥ E [log2 det (IrA + SINRHuH∗u)] (2.8)

where

SINR =(P/tA) /Lu

N + (P/tA)∑v 6=u 1/Lv

(2.9)

The SINR in the above equation can be seen as the Signal to Interference and Noise Ratioper transmitting antenna.

Proof. Let E [·|Hu] designate the expectation conditionally to Hu. By the properties of theconditional expectation we have

E [C] = E [E [C|Hu]]

Equation (2.4) implies that

E [C|Hu] ≥ E[

log2 det

(IrA +

P

tA

HuH∗u

Lu∆−1

)∣∣∣∣Hu

]where ∆ is given by (2.3).

Using Jensen’s inequality and convexity of the function A 7→ log2 det(IrA + P

tAHH∗A−1

)on

the set of positive definite matrices of CrA×rA (cf. [40, Lemma II.3]), we deduce that

E [C|Hu] ≥ E [ log2 det (IrA + SINRHuH∗u)|Hu]

where the SINR is given by (2.9). Thus

E [C] = E [E [C|Hu]]

≥ E [E [log2 (1 + SINRHuH∗u) |Hu]]

= E [log2 (1 + SINRHuH∗u)]

Remark 2 Note that the right-hand side of (2.8) represents the capacity of a MIMO channel withAWGN channel and i.i.d. circularly symmetric Gaussian fading given in Telatar [91, Theorem 1].Thus, it may be calculated using the analytic formula given in [91, Theorem 2] or approximatedwith the help of the asymptotic result of Lemma 4 stated in the Appendix as follows

E [C] ≥ E [log2 det (IrA + SINRHuH∗u)]

' rAlog (2)

C(tA × SINR,

tArA

)(2.10)

where C is given by (2.30).

2.4. MMSE 23

Remark 3 Time/frequency diversity. Averaging over a large number of time-slots correspondsto exploiting the so-called time-diversity, which is suitable for the analysis of the performanceof a variable bit-rate traffic as observed in [30, §I]. Consider now a given time-slot and largenumber n of sub-carriers. Assume that the fadings for different sub-carriers are i.i.d., thenagain, by the law of large numbers, the capacity of a large number n of sub-carriers approachesthe ergodic capacity. Thus the ergodic capacity is also appropriate for a constant bit-rate trafficprovided the number of sub-carriers allocated to each user is large enough. If the number of sub-carriers allocated to each user is not sufficiently large, then, as observed in [30, §I], a relevantperformance indicator is the outage probability, defined as the probability that the capacity in agiven time-slot is smaller than the desired bit-rate r. Evaluation of this latter characteristic isnot in the scope of this thesis.

Comparison of the lower bounds

We aim now to compare numerically the bounds (2.7) and (2.8). In this regard, we consider ahexagonal cell surrounded by 6 neighboring base stations. The distance between two base stationsis 0.5km and the distance propagation law, i.e. path-loss is l (r) = (Kr)

βwhere K = 7764,

β = 3.52 which are the typical values in urban areas. We consider that a noise power equalsN = −93dBm, standard deviation of shadowing of 8dB and a transmission power of the basestation P = 58.5dBm. We consider 2 receiving antennas and a number of transmitting antennastA ∈ 1, 2, 8. Figure 2.1 gives the capacity lower bounds (2.7) and (2.8) called, respectively,asymptotic and Jensen bound, as function of the distance between the user and the central basestation. This figure shows that the two bounds are close to each other.

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2 0.25 0.3

Cap

acity

per

rec

eivi

ng a

nt. [

nats

/s/H

z]

Distance [km]

Asymptotic bound, 1x2Jensen bound




Figure 2.1: Jensen (2.8) and asymptotic (2.7) lower bounds for the capacity (peak bit-rate) of theMIMO flat fading channel with additive noise in the downlink of an OFDMA cellular network.Capacity as function of the distance between the user and his serving base station.

2.4 MMSE

The linear MMSE (Minimum Mean Square Error) decoder of the channel (2.1) means that thereceiver estimates the transmitted signal Ψn using a linear transformation Ψn of the received

signal Yn minimizing the error E

[∥∥∥Ψn − Ψn

∥∥∥2]. MMSE-SIC (Successive Interference Cancella-


tion) consists in decoding successively the tA transmitting antennas while suppressing recursivelythe previously decoded signals. The objective of the present section is to establish lower boundsfor the capacity of MMSE and MMSE-SIC for a cellular network where interference is neitherwhite nor Gaussian.

2.4.1 MMSE capacity given fading

We consider a single time-slot and sub-carrier in this section; thus the fading is assumed given.It follows from the general theory of linear estimation [28, §3.3] that

Ψn = ΓΨnYnΓ−1YnYn, n ∈ N (2.11)

where ΓYn = E [YnY∗n ] = ΓΣ+ P

tALuHH∗ is the covariance matrix of Yn and ΓΨnYn = E [ΨnY

∗n ] =

P

tAL1/2u

H∗ is the covariance matrix of Ψn and Yn. Equation (2.11) is in fact a system of tA

equations corresponding to the estimation of the signals emitted by the different transmittingantennas of the serving BS. More specifically, denoting by Ψn (k) the signal emitted by the k-thantenna and Ψn (k) the corresponding estimation, Equation (2.11) decomposes into

Ψn (k) = ΓΨn(k)YnΓ−1YnYn, n ∈ N

where ΓΨn(k)Yn = E [Ψn (k)Y ∗n ]. The above equation may be written in the form

Ψn (k) = α∗Ψn (k) + zn

where α ∈ C and zn is a random variable with values in C. The above expression may be seen asthe input-output relation of an additive channel corresponding to the k-th transmitting antenna.It is shown in [92, Equation (8.67)] that the corresponding signal to noise power ratio equals

SNRk =E[|α∗Ψn (k)|2

]E[|zn|2

] =P

tALuh∗kΓ−1

k hk, k = 1, . . . , tA (2.12)

where hk is the k-th column of the fading matrix Hu and

Γk = ∆ +

tA∑i=1,i6=k

P

tALuhih∗i , k = 1, . . . , tA

where ∆ is given by (2.3). It follows from Corollary 2 that the capacity of the k-th transmittingantenna (when considering interference from other antennas as well as from other BS) is lowerbounded by

Ck ≥ log2 (1 + SNRk) , k = 1, . . . , tA

Thus the capacity of the channel is lower bounded by

CMMSE =

tA∑k=1

Ck ≥tA∑k=1

log2 (1 + SNRk) (2.13)

2.4. MMSE 25

2.4.2 MMSE ergodic capacity

Consider now a given sub-carrier and multiple time-slots. The capacity of the channel is thenthe expectation of the above capacity (2.13) with respect to fading; thus

E [CMMSE] ≥tA∑k=1

E [log2 (1 + SNRk)]

=

tA∑k=1

E [E [ log2 (1 + SNRk)|Hu]]

Using [40, Lemma II.3] and Jensen’s inequality, it follows that

E [CMMSE] ≥tA∑k=1

E

[log2

(1 +

P

tALuh∗kΓ−1

k hk

)](2.14)

where

Γk = E [Γk|Hu] =

N +P

tA

∑v 6=u

1

Lv

IrA +

tA∑i=1,i6=k

P

tALuhih∗i

The right-hand side of (2.14) may be evaluated numerically using Monte Carlo method based onsamples of the fading matrix Hu.

2.4.3 MMSE-SIC

As we said previously, MMSE-SIC consists in decoding successively the tA transmitting antennas,but before decoding the signal from a given antenna we suppress the previously decoded signals.Thus the channel for the k-th transmitting antenna is an additive channel with SNR givenby (2.12) where the matrix Γk is now given by

Γk = ∆ +

tA∑i=k+1

P

tALuhih∗i , k = 1, . . . , tA

The lower bound (2.13) of capacity given the fading remains valid with the above modificationof SNR. The lower bound (2.14) of the ergodic capacity holds also true with

Γk =

N +P

tA

∑v 6=u

1

Lv

IrA +

tA∑i=k+1

P

tALuhih∗i

The proof is based on Jensen inequality and follows the same lines as for MMSE. It followsfrom [92, Equation (8.71)] that

tA∑k=1

E

[log2

(1 +

P

tALuh∗kΓ−1

k hk

)]= E [log2 det (IrA + SINRHuH

∗u)]

where SINR is given by (2.9). Note that the right-hand sides of the above equation and Equa-tion (2.8) are equal; that is we retrieve the same capacity lower bound as for the original chan-nel (2.1).


2.5 Numerical results for the link capacity

The objective of the present section is to compare the theoretical expressions established in theprevious section to real field measurements and to some simulation compliant with the 3GPPrecommendation [3].

2.5.1 Link layer model calibration

We consider firstly a user served by a base station through an additive white Gaussian noise(AWGN) SISO channel neglecting fading and interference for the moment. The user gets ideally(i.e. in the asymptotic sense of information theory) a bit-rate given by the famous Shannon’sformula w log2 (1 + SNR) where w is the bandwidth allocated to the considered user and SNRis the signal to noise power ratio. In order to get rid of the dependence of the bit-rate on thebandwidth, we define the spectral efficiency as the ratio of the bit-rate to the bandwidth whichequals log2 (1 + SNR) in the AWGN context.

Mogensen et al. [73], [53] and the 3GPP [4, §A.2] have observed that the LTE system spectralefficiency in this AWGN context is well approximated by

C ' c log2 (1 + qSNR) (2.15)

for some constant c < 1 and q accounting on the one hand for the gap between the practicalcoding schemes and the optimal ones and on the other hand for the loss of capacity due tosignalling. This observation shall be confirmed and the typical value of c and q for LTE will begiven.

First, we will calibrate these parameters c and q for real coding schemes considering thesimplest AWGN SISO channel, and then use them in the analysis of the MIMO channel withfading and interference.

Note that the relative difference 1−c for q = 1 between the Shannon’s limit and the practicalLTE system may be seen as a progress margin for potential evolution of the technology in theAWGN context.

According to [2, §6.8], [35, p.155] LTE signalling consumes about 30% of the available capacity.On the other hand, different M -QAM modulations with M ∈ 4, 16, 64 are used with linkadaptation and a target block error rate 10−2. Moreover, CRC and turbo coding are implemented.The 3GPP [4, §A.2] shows that the bit-rate of LTE is about 25% smaller than the Shannoncapacity. In order to account for these losses, we assume

c = (1− 0.3)× (1− 0.25) ' 0.5, and q = 1 (2.16)

in Equation (2.15).

Figure 2.2 shows that the analytic approximation (2.15) of the SISO capacity with the valuesof c and q proposed in (2.16) fits well the results of Orange’s link simulation tool in AWGN.Thus we will retain c = 0.5 and q = 1 to weigh, respectively, the capacity and the SINR in thesubsequent analysis of the MIMO channel with fading and interference.

More specifically, using the modified AWGN formula (2.15), the capacity lower bound (2.10)becomes

E [C] ≥ c rAlog (2)

C(tA × SINR,

tArA

)(2.17)

where the parameter c is given by (2.16) and the function C is given by Equation (2.30).

2.5. NUMERICAL RESULTS FOR THE LINK CAPACITY 27

0

0.5

1

1.5

2

2.5

3

3.5

-5 0 5 10 15 20

Cap

acity

per

uni

t ban

dwid

th [b

it/s/

Hz]

SNR [dB]

SimulationsAnalytic: a=0.5, q=1

Figure 2.2: Performance of SISO without fading evaluated using analytic approximation (2.15)and link simulation.

2.5.2 Comparison to simulation

We compare now our theoretical bound to 3GPP simulation [3]. The simulation is carried outunder the following assumptions. There are 2 transmitting and 2 receiving antennas. Eachbase station always transmits at its maximal power. The receiver is MMSE-SIC (interferencecancellation), the channel is 3GPP Spatial Channel Model (SCM) and users’ speed is 3km/h.Several realizations of the user positions, shadowing losses and fading channels are generated.For each user location and each shadowing realization, the capacity is averaged over about 1000fadings samples. Moreover, the value of the SINR including only the distance and the shadowingeffects (and not fading) is also given. The simulation accounts for correlations between individualMIMO sub-antennas.

Figure 2.3 shows the simulation results compared to the analytic bound for MMSE-SIC withcorrelated antennas. Observe that the analytic curves agree with simulation results; but there ismore variability in simulation due to the averaging over fading which does not yet converge tothe ergodic capacity.

2.5.3 Comparison to measurements

We take measurements from the city of Marseille. These measurements are collected by dedicatedusers in the downlink of Orange’s experimental LTE network composed of 75 cells each having2 transmitting antennas. The mobiles used for measurements have also 2 receiving antennas.Carrier frequency is 2.6GHz, bandwidth is 20MHz.

Figure 2.4 shows these measurements compared to the analytic bounds (2.14) and (2.8) forMMSE and MMSE-SIC respectively.

The curve ’MMSE Correl’ shows the results of the MMSE scheme with correlations betweenindividual MIMO antennas; more precisely we assume a correlation factor of 0.3 for transmittingantennas and 0.9 for receiving antennas as proposed by 3GPP [5, §B.2.3]. We observe that thisassumption fits the real performance of the current network. The curve ’MMSE-SIC Correl’predicts the performance of the MMSE-SIC scheme still with correlated antennas. If technol-ogy allows for decorrelated antennas, then the performance can reach the values predicted by”MMSE Uncorr’ and ’MMSE-SIC Uncorr’ depending on the used scheme. Recall that MMSE-SIC with decorrelated antennas gives the full MIMO capacity and that the corresponding analytic


0

2

4

6

8

10

12

14

-10 -5 0 5 10 15 20

Spe

ctra

l effi

cien

cy [b

it/s/

Hz]

SINR [dB]

3GPP simulationsAnalytical

Figure 2.3: Analytic relation of the peak bit-rate to SINR compared to 3GPP simulation

bound (2.8) is well approximated by the asymptotic expression (2.17) as shown in Section 2.3.3.

2.5.4 Approximate link quality estimation via simulations

The goal is to use the Orange’s internal 3GPP simulators mentioned earlier and develop a quickand simple estimation of link performance for different configurations of MIMO. Indeed thesignalling loss depends on the number of transmitting and receiving antennas and consequentlythe weighting constants c and q. It is about 40/168 = 24%, 48/168 = 29% and 52/168 = 31%respectively for SIMO 1× 2, MIMO 2× 2 and MIMO 4× 2 (see [2, §6.8], [35, p.155]). Here, allthe considered cases have a MRC (Maximum Ratio Combining) receiver, except the MIMO 4×2case which has a MMSE receiver, which is different compared to Section 2.5.2, where MMSE-SICis used. At the base station side, the transmitting antennas are pairwise cross-polar. In the caseMIMO 4 × 2 the two cross-polar pairs of transmitting antennas are separated by 10 times thewavelength.

In order to simplify the notation, we denote by S the analytical (lower bound for the) spectralefficiency given in the right-hand side of (2.8) weighted by the parameter c = 0.5 obtained in theprevious section; that is

S (SINR, tA, rA) = cE [log2 det (IrA +HuH∗uSINR)] (2.18)

where SINR is the signal to interference and noise ratio (per transmitting antenna) given byEquation (2.9).

In order to get the practical LTE performance, we make the same kind of comparison as inSection 2.5.2. We consider the output of Orange’s simulator compliant with the 3GPP recom-mendation [3] (see this reference for the details of the simulations) in the so-called calibrationcase. It corresponds to MIMO 1 × 2 with round robin (RR) scheduler. We consider also otherMIMO configurations and proportional fair (PF) scheduler, keeping all the other parametersunchanged. In particular, each base station always transmits at its maximal power (full buffer).

The spectral efficiency as function of the SINR is compared to the theoretical relation (2.18) orequivalently (2.17). More specifically, we make a linear regression between the spectral efficiency

2.5. NUMERICAL RESULTS FOR THE LINK CAPACITY 29

0

2

4

6

8

10

12

14

16

18

20

-15 -10 -5 0 5 10 15 20 25 30

Pea

k bi

t-ra

te p

er u

nit b

andw

idth

[bps

/Hz]

SINR [dB]

3GPP SimulationMMSE-SIC UncorrMMSE-SIC Correl

MMSE UncorrMMSE Correl

Figure 2.4: Analytic relations of the peak bit-rate to SINR compared to measurements; seeSection 2.5.3 for the details.

MIMO Scheduler b residual stand. dev. b′

1× 2 RR 0.83 0.45 0.981× 2 PF 1.02 0.65 1.192× 2 PF 0.67 0.74 1.084× 2 PF 0.49 0.76 0.90

Table 2.1: Results of the linear fittings.

obtained from simulations and the theoretical efficiency given by Equation (2.18); that is wesearch for some b such that

s ' b× S (SINR, tA, rA) (2.19)

Table 2.1 gives the results of the linear fitting (2.19); i.e. the values of b and the correspondingresidual standard deviation for different MIMO configurations (the first row corresponds to thecalibration case [3, Table A.2.2-1]). Moreover, the 95%-confidence interval is about b± 0.01 forall the studied cases.

Figure 2.5 shows the spectral efficiency as function of the SINR from simulations and fromthe analytical expression (right-hand side of (2.19)) for the calibration case. Observe again thatthe analytical expression reproduces well the general tendency of the empirical data obtainedfrom simulations, similar as in Section 2.5.2.

Remark 4 In order to simplify the calculations we have also tested a linear regression betweenthe spectral efficiency s obtained from simulations and the AWGN expression (2.15). Observefrom Equation (2.9) that when noise is dominant against interference, then

SINR =(P/tA) /LuN

=P/LuN

× 1

tA

Thus, in this particular case, the term P/LuN in the right-hand side of (2.15) equals SINR × tA.

Then, in the general case, it is natural to look for a fitting in the form

s ' b′ × c log2 (1 + SINR× tA)

The resulting values of b′ are indicated in Table 2.1 with residual standard deviations close tothose indicated in the fourth column of that table.


0

1

2

3

4

5

6

-10 -5 0 5 10 15 20

Spe

ctra

l effi

cien

cy [b

it/s/

Hz]

SINR [dB]

3GPP simulationsAnalytical

Figure 2.5: Simulations versus the analytical expression (right-hand side of (2.19)) for the cali-bration case

2.6 Link quality observed by a typical user

The objective of the present section is to estimate the spatial distribution of the link performanceparameters (SINR, spectral efficiency etc.) observed by a typical user whose location will berandomly chosen in the network. We generate the link performance parameters distributionbased on the aforementioned study and in the context of the 3GPP simulation scenario [3]. So,we want to apply the above analysis to produce for example the spatial distribution of the SINR,and in such a manner provide the link quality ingredient as a corner stone in our further Qualityof Service (QoS) examination.

2.6.1 SINR

For the analytical approach we use a similar geometric pattern of the network (hexagonal) and thesame propagation-loss modeling regarding the distance and shadowing effects (fading has beenalready taken into account on the link level in the previous section) as the 3GPP calibrationcase [1, Table A.2.1.1-3] and [3, Table A.2.2-1].

More specifically, the frequency carrier is 2GHz. The path-loss model is l(r) = 128.1 + 37.6×log10(r) [in dB]. A supplementary penetration loss of 20dB is added. The shadowing is modeledas a centered log-normal random variable of standard deviation 8dB. The following 2D horizontalantenna pattern is used

A (ϕ) = −min

(12

(ϕ

ζ

)2

, Am

), ζ = 70, Am = 20dB (2.20)

The system bandwidth is W = 10MHz, the noise power equals N = −95dBm (−174dBm/Hz,noise figure=9dB) and the transmission power of the base station is P = 60dBm (46dBm plusG = 14dBi of antenna gain). The network is composed of 36 hexagons (6× 6). Each hexagoncomprises three sectors which gives a total of 108 sectors. The distance between the centers of

2.6. LINK QUALITY OBSERVED BY A TYPICAL USER 31

two neighboring hexagons is 500m. We generate 3600 random user locations uniformly in thenetwork; that is 100 user locations per hexagon on average.

The 3GPP simulations published in [3] are made on a planar network with random locationsof the users. In the present study, two network models are considered: either planar or toroidal(to avoid the border effects).

Each mobile is served by the base station with the smallest propagation-loss (including dis-tance, shadowing and antenna pattern). In order to facilitate the comparison of our results tothose of 3GPP, we define the coupling-gain as the antenna gain G minus propagation-loss L withthe serving base station. The cumulative distribution function (CDF)2 of the coupling gain ob-tained by 3GPP simulations [3, Figures A.2.2-1 (left)] and by our models are given in Figure 2.6.This figure shows that the results of our planar network are close to those of 3GPP simulations,whereas those of the toroidal network give larger coupling gain. This is due to the fact that ina planar network edge users get smaller coupling gain than in the toroidal one.

0

0.2

0.4

0.6

0.8

1

-140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40

CD

F

Coupling gain [dB]

3GPP simulationsPlanar network

Toroidal network

Figure 2.6: CDF of the coupling gain (antenna gain minus propagation loss)

The SINR for each mobile is calculated by Equation (2.9), where u is the index of the servingbase station. Figure 2.7 shows the CDF of the SINR coming from 3GPP simulations [3, FigureA.2.2-1 (right)] compared to that resulting from our models. Again our planar model gives closerresults to the 3GPP simulations than the toroidal one. Nevertheless, the difference between theSINRs of the toroidal and the planar networks is smaller than 0.5dB.

Figure 2.7 shows that the SINR does not exceed 17dB. Indeed, each mobile served by a givenbase station (sector) is at least interfered by the two other sectors on the same site. The powerreceived from each of these sectors is at least 10−2 times that received from the serving BS (thisis related to Am = 20dB in Equation (2.20)). The interference to signal ratio is consequentlylarger than 2× 10−2 i.e. −17dB which explains the observed upper limit of the SINR.

Remark 5 Observe that the SINR defined by Equation (2.9) is different from the SINR calcu-

2over all the user locations in the network


0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10 15 20

CD

F

SINR [dB]


Toroidal network

Figure 2.7: CDF of SINR

lated by 3GPP simulations which equals

SINR3GPP =P/Lu

N + P∑v 6=u 1/Lv

However, if noise is negligible compared to interference, then the two SINRs are identical. Thisis the case in the considered urban scenario (small cell radius), so we do not have to distinguishbetween these two SINRs.

2.6.2 Spectral efficiency

For each mobile we calculate the spectral efficiency corresponding to its SINR by relation (2.19).In order to facilitate the comparison of our results to those of 3GPP, we define the normalized userthroughput as the spectral efficiency divided by 10 (this is historically related to the fact there are10 users per cell in 3GPP simulations). The CDFs of the normalized user throughput obtained by3GPP simulations [3, Figure A.2.2-3 (left)] and by our model are plotted in Figure 2.8. The 3GPPdistribution is more spread than that of our models; this is related to the fact that the 3GPPspectral efficiency represents some variability around the analytic one as shown in Figure 2.5.Moreover, we observe that the results of the planar and toroidal models for the network are closeto each other. Thus, the toroidal model is considered for the remaining part of the Section.

Table 2.2 gives the arithmetic mean of the spectral efficiencies at the different locations (calledcell spectral efficiency) for both, the 3GPP simulations and analytic approach. The results oftwo methods agree for all the considered MIMO and scheduler configurations.

Note that the results of the simulations given in Table 2.2 are produced by the simulator ofOrange which is one of the contributors to 3GPP. The values indicated in [3, Table A.2.2-2] arein fact averaged over the different 3GPP contributors including Orange. In particular, for thecalibration case (MIMO 1× 2 with RR scheduler) Orange’s result is 1.01 whereas 3GPP averageis 1.1. The variability of the results amoung the contributors is partially due to the randomnessinduced by the shadowing.

2.6. LINK QUALITY OBSERVED BY A TYPICAL USER 33

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

CD

F

Normalized user throughput [bit/s/Hz]


Toroidal network

Figure 2.8: CDF of normalized user throughput

Arithmetic mean Harmonic mean

MIMO Scheduler Simus Analytic Simus Analytic

1× 2 RR 1.01 1.00 0.50 0.691× 2 PF 1.32 1.23 0.80 0.852× 2 PF 1.43 1.41 0.84 1.004× 2 PF 1.54 1.54 0.95 1.18

Table 2.2: Cell spectral efficiency: Comparison of the 3GPP simulations and the analytic results.

Table 2.2 shows the harmonic means of the spectral efficiency obtained from 3GPP simulationsand from the analytical expression. The difference may be explained as follows. Recall that theharmonic mean is sensitive to the minimal value of the considered data; for example if one ofthese data is null then the harmonic mean vanishes. Moreover, Figure 2.5 shows that the 3GPPspectral efficiency represents some variability around (and in particular comprise smaller valuesthan) the analytic curve. This explains why the harmonic means obtained from simulations inTable 2.2 are lower than the analytic ones.

Appendix

2.A Theoretical results: MIMO flat-fading channel withadditive noise

We shall establish in what follows a useful lower bound of the capacity of a general additivenoise channel, where the noise is not necessarily Gaussian nor white. Our motivation is thatinterference in wireless networks does not have necessarily these properties.

2.1.1 Model

Consider a discrete-time model of a multiple input and multiple output (MIMO) channel with tAtransmitting and rA receiving antennas such that, at each time n = 1, 2, . . ., the channel outputYn ∈ CrA is related to the channel input Ψn ∈ CtA by

Yn = HΨn + Σn, n = 1, 2, . . . (2.21)

where H is a complex matrix of dimension rA × tA modelling the fading, and Σ1,Σ2, . . . ∈ CrAis the noise process. We assume that the fading matrix H is deterministic. The channel input issubject to a power constraint of the form

1

n

n∑k=1

Ψ∗kΨk ≤ P, n = 1, 2, . . .

where P is a given positive constant and Ψ∗k designates the transpose complex conjugate of Ψk.Note that the above constraint concerns the total power aggregated over all the tA transmittersand averaged over n channel uses. The channel (2.21) is called MIMO additive noise channelwith deterministic fading.

2.1.2 Capacity lower bound

We are interested in the capacity of the channel (2.21) when the noise samples Σ1,Σ2, . . . arenot necessarily Gaussian nor independent. We shall in fact establish an explicit lower bound forthis capacity.

We begin by some definitions and notation. The identity matrix of dimension rA × rA isdenoted by IrA . The covariance matrix of a centred random vector Ψ ∈ CtA is denoted by

ΓΨ = E [ΨΨ∗]

The covariance matrix of two centred random vectors Ψ ∈ CtA and Y ∈ CrA is denoted by

ΓΨY = E [ΨY ∗]

35


A random vector Ψ ∈ Cn is called circularly-symmetric if eiφΨ has the same distribution as Ψfor all φ ∈ R which implies that Ψ is centred.

From now on all the considered random vectors are assumed to have well defined entropies [56,§ 1.3]. For example, if the random vector Ψ ∈ Cn has a density pΨ with respect to the Lebesguemeasure on Cn, then its entropy is defined by h(Ψ) = −

∫Cn pΨ(x) log pΨ(x)dx provided the

Lebesgue integral is well defined. We denote by I (Ψ;Y ) the mutual information between tworandom vectors Ψ and Y which is related to the entropy by [56, Theorem 1.6.2]

I (Ψ;Y ) = h (Ψ)− h (Ψ|Y ) (2.22)

where h (Ψ|Y ) is the entropy of Ψ conditionally to Y .We give now two preliminary lemmas.

Lemma 2 Let Ψ1,Ψ2, . . . ,Ψn be random vectors in CtA and Y1, Y2, . . . , Yn be random vectorsin CrA . Denote Ψ(n) = (Ψ1,Ψ2, . . . ,Ψn) and Y (n) = (Y1, Y2, . . . , Yn). If Ψ1,Ψ2, . . . ,Ψn areindependent, then

I(

Ψ(n);Y (n))≥

n∑k=1

I (Ψk;Yk)

Proof. The mutual information may be expressed in terms of the entropy as follows [56,Theorem 1.6.2]

I(

Ψ(n);Y (n))

= h(

Ψ(n))− h

(Ψ(n)|Y (n)

)Since Ψ1,Ψ2, . . . ,Ψn are independent, the entropy h

(Ψ(n)

)may be decomposed as the sum of

the individual entropies [56, Theorem 1.3.2 (h.6)] h(Ψ(n)

)=∑nk=1 h (Ψk). On the other hand,

the conditional entropy h(Ψ(n)|Y (n)

)may be bounded as follows

h(

Ψ(n)|Y (n))

=

n∑k=1

h(

Ψk|Y (n),Ψ1, . . . ,Ψk−1

)≤

n∑k=1

h(

Ψk|Y (n))≤

n∑k=1

h (Ψk|Yk)

where for the first equality we use [56, Theorem 1.3.2 (h.7)] and for the two above inequalitieswe use [56, Theorem 1.3.2 (h.7)] and [56, Theorem 1.3.2 (h.5)] respectively. Combining the abovethree equations, we get the desired result.

The following lemma may be seen as an extension of [56, Theorem 1.8.6] or [40, Lemma II.2]to the complex case. Our proof is inspired by [71] and [64].

Lemma 3 Consider three random vectors Ψ ∈ CtA , Y, Y ∈ CrA . Assume that the random vector(Ψ, Y

)is circularly-symmetric Gaussian with the same covariance matrix as (Ψ, Y ) and that

ΓY is invertible. ThenI(Ψ;Y ) ≥ I(Ψ; Y )

Proof. For any deterministic matrix A ∈ CtA×rA ,

h (Ψ|Y ) = h (Ψ−AY |Y ) ≤ h (Ψ−AY ) (2.23)

where for the above inequality we use [56, Theorem 1.3.2]. In particular, taking A = ΓΨY Γ−1Y in

which case AY is the best quadratic approximation of Ψ by a linear function of Y , and lettingU = Ψ−AY , we get

h (Ψ−AY ) = h(U) ≤ log [det (πeΓU )] (2.24)

2.A. THEORETICAL RESULTS 37

Combining the above two inequalities we get h (Ψ|Y ) ≤ log [det (πeΓU )]. Then Equation (2.22)implies

I (Ψ;Y ) ≥ h (Ψ)− log det (πeΓU ) (2.25)

Apply now the above arguments with Y in the role of Y . Observe that U = Ψ − AYis circularly-symmetric Gaussian, thus equality holds in (2.24). Moreover, U is independent

from Y since ΓUY = E[U Y ∗

]= ΓΨY − AΓY = 0, and decorrelation implies independence for

circularly-symmetric Gaussian random vectors. Thus equality holds also in (2.23) which showsthat

I(

Ψ; Y)

= h (Ψ)− log det (πeΓU )

which combined with the observation that ΓU = ΓΨ−ΓΨY Γ−1Y ΓYΨ = ΓU and (2.22) finishes the

proof of the desired inequality.We show now that the above lemmas permit to deduce a lower bound for the capacity of

the channel (2.21). The considered channel has memory; i.e. different channels uses are notindependent because of the noise samples might be correlated, thus its information capacity Cis defined as follows

C = lim infn→∞

1

nC(n)

where

C(n) = supΨ(n)

I(

Ψ(n);Y (n))

;1

n

n∑k=1

E [Ψ∗kΨk] ≤ P

where Ψ(n) = (Ψ1, . . . ,Ψn) is a random object with values in (CtA)

n; and Y (n) is the output of

the channel associated to the input Ψ(n).

Proposition 3 Assume that the covariance matrix E[ΣkΣ∗k] of the noise Σk is finite for allk ∈ N and denote

∆n =1

n

n∑k=1

E[ΣkΣ∗k]

Then the information capacity of the channel (2.21), given the fading matrix H, is lower boundedby

C ≥ lim infn→∞

[log2 det

(IrA +

P

tAHH∗∆−1

n

)](2.26)

The above inequality remains true under the additional constraint that the signals emitted by thetransmitting antennas are independent and have equal powers.

Proof. Consider independent inputs Ψ1,Ψ2, . . ., then by Lemma 2

I(

Ψ(n);Y (n))≥

n∑k=1

I (Ψk;Yk)

Assume now that each Ψk is circularly-symmetric Gaussian, independent of Σk, and with covari-ance matrix E[ΨkΨ∗k] = P

tAItA . Note that

E [ΨkY∗k ] = E [ΨkΨ∗kH

∗] + E[ΨkΣ∗k] =P

tAH∗

and

E [YkY∗k ] = E [HΨkΨ∗kH

∗] + E[ΣkΣ∗k] =P

tAHH∗ + E[ΣkΣ∗k]


Consider Σk circularly-symmetric Gaussian independent from Ψk and with the same covariance

matrix as Σk. Denoting Yk = HΨk + Σk, then(

Ψk, Yk

)is circularly-symmetric Gaussian with

the same covariance matrix as (Ψk, Yk). We deduce from Lemma 3 that

I(Ψk;Yk) ≥ I(Ψk; Yk)

= h(Yk)− h(Yk|Ψk)

= h(Yk)− h(Σk)

= log2

[det

(πe

(P

tAHH∗ + E[ΣkΣ∗k]

))]− log2 [det (πeE[ΣkΣ∗k])]

= log2 det

(IrA +

P

tAHH∗E[ΣkΣ∗k]−1

)Using Jensen’s inequality and convexity of the function A 7→ log2 det

(IrA + P

tAHH∗A−1

)on

the set of positive definite matrices of CrA×rA , cf. [40, Lemma II.3], we obtain

1

n

n∑k=1

I (Ψk;Yk) ≥ log2 det

(IrA +

P

tAHH∗∆−1

n

)which concludes the proof of (2.26).

The last statement in the Proposition follows from the fact that the above inequality is provedfor inputs Ψ1,Ψ2, . . . such that each Ψk is circularly-symmetric Gaussian with covariance matrixE[ΨkΨ∗k] = P

tAItA , which implies that, for each k ∈ N∗, the components of the vector Ψk are

independent from each other.We make an observation and give a corollary.

Remark 6 Assume that Σ1,Σ2, . . . have the same covariance matrix E[ΣkΣ∗k] = ∆, then, theright-hand side of (2.26) equals

log2 det

(IrA +

P

tAHH∗∆−1

)Note that the above formula gives also the capacity of a MIMO channel with additive circularly-symmetric Gaussian noise process with independent samples and equi-partition of power betweenthe transmitting antennas.

The following Corollary of Proposition 3 states that, for a single input and single output(SISO) channel tA = rA = 1, the worst additive noise process distribution (not necessarily whitenor Gaussian) for capacity with given second moment, is the additive white Gaussian noise(AWGN). This result may be seen as an extension of Gallager’s result [47, Theorem 7.4.3] formemoryless channels to the channels with memory. It may also be deduced from Shannon’sresult [84, Theorem 18]–proved there by the entropy power inequality– and from the fact thatthe entropy power is not larger than the average power.

Corollary 2 Consider a SISO channel whose input and output, at time n, represented by Ψn ∈C, Yn ∈ C respectively, are related by

Yn = Ψn + Σn, n = 1, 2, . . .

2.A. THEORETICAL RESULTS 39

where the noise process Σ1,Σ2, . . . ∈ C is assumed stationary and satisfies E[|Σn|2

]= N .

Assume that the channel has a power constraint in the form 1n

∑nk=1 |Ψk|2 ≤ P . Then the

information capacity C of the channel is lower bounded by

C ≥ log2

(1 +

P

N

)(2.27)

2.1.3 Asymptotic analysis

Consider a particular case where the noise samples Σ1,Σ2, . . . are i.i.d. each being circularly-symmetric Gaussian with covariance matrix E[ΣnΣ∗n] = N IrA where N is a given positiveconstant. In this case, the right-hand side of (2.26) equals

log2 det

(IrA +

P

tANHH∗

)= log2 det

(IrA +

P

tAHH∗

)(2.28)

where P = PN is the signal to noise power ratio (SNR). For given tA and rA the capacity (2.28)

depends on H.Frequently, one is interested in the ergodic capacity, that is the expectation of the capacity

with respect to the fading matrix H assumed random with a given distribution. Assume forexample that H has i.i.d. components each being circularly-symmetric Gaussian with variance

1. In this case, the expectation E[log2 det

(IrA + P

tAHH∗

)]may be calculated with the help of

the analytical result given by Telatar [91, Theorem 2].Alternatively, the capacity (2.28) may be approximated with the help of the following asymp-

totic result saying that when the number of transmitting and receiving antennas go to infinity,the capacity per receiving antenna converges to a deterministic limit.

0

1

2

3

4

5

6

7

-15 -10 -5 0 5 10 15 20 25 30

Cap

acity

per

rec

eivi

ng a

nt. [

nats

/s/H

z]

SNR [dB]

Analytic, 1x2Asymptotic




Figure 2.9: Comparison of the analytic capacity and the asymptotic formula

Lemma 4 [95, Equations (9), (38)], [79, Appendix] Assume that the fading matrix H ∈ CrA×tAhas i.i.d. components, centred and with variance 1. Assume that tA, rA → ∞ such that tA

rA→

Q ∈ R+, then1

rAlog det

(IrA +

P

tAHH∗

)→ C (P,Q) (2.29)


almost surely, where

C (P,Q) := Q log

(1 +

P

Q− 1

4F(P

Q,Q

))+ log

(1 + P − 1

4F(P

Q,Q

))− Q

4PF(P

Q,Q

)(2.30)

where

F (ξ,Q) =

(√ξ(

1 +√Q)2

+ 1−√ξ(

1−√Q)2

+ 1

)2

The above asymptotic result gives a good approximation of the expectation E[

1rA

log det(IrA + P

tAHH∗

)]even for a small number of antennas as already observed in [69, Table 1] for SNR = 10dB andconfirmed for SNR ∈ [−15dB, 30dB] in Figure 2.9 where we consider 2 receiving antennas and anumber of transmitting antennas tA ∈ 1, 2, 4, 8.

Chapter 3

User throughput versus trafficdemand — global networkperformance via a fixed pointproblem

3.1 Introduction

The traffic demand in wireless cellular networks is increasing rapidly and is expected to doubleevery year. To respond efficiently to this demand the new generation of mobile cellular systemscalled LTE is being developed as a successor of the currently deployed 2G and 3G systems.

The deployment of such networks, frequently based on coverage conditions, should now berevised to account for this traffic increase. In particular, a densification is sometimes required.But how many sites are required to satisfy a given traffic demand with a specified quality ofservice target? This is a dimensioning problem. Another important question is to establisha relationship between the QoS perceived by the users, e.g. user throughput and the trafficdemand. This would enable for example network operators to know how close their networks areto some ”stability” limits. We focus on variable bit-rate (VBR) traffic; that is users requiringsome volume of date to transmit at a bit-rate which may be decided by the network. In thiscase, the traffic demand may be expressed in bit/s/km2 and the quality of service in terms ofthe mean throughput (in bit/s) offered to users in the long run of arrivals and departures.

The objective of this chapter is to develop an approach based on queueing theory and stochas-tic geometry, as well as on the previous results of Chapter 2, to tackle the dimensioning and QoSprediction problems in an efficient way.

In the present chapter we continue building the analytical approach and consider two networkscenarios: symmetric (regarding the spatial pattern of base stations’ positions) or regular andnon-symmetric or irregular:

• We account for the dynamics of call arrivals and departures and calculate within thiscontext the QoS perceived by users. This represents a step forward compared to theclassical coverage (or static capacity) point of view.

• We continue the idea proposed in [82] of studying the dependence of load on traffic demand

41

42 CHAPTER 3. USER THROUGHPUT

by developing an analytical approach based on queueing theory.

• We show and mathematically capture the dependence between loads of base stations in thenetwork via a fixed-point problem.

• We use the obtained results to show how to dimension a symmetric network.

• Exploring the results of Chapter 2 and a queueing approach analysis we apply the resultsof stochastic geometry and examine the user QoS in non-symmetric network e.g. Poisson,which is more realistic scenario.

• We show how the mean user throughput depends on the traffic demand in large irregularnetworks and after give the spatial CDF of the following QoS parameters: cell load, meanusers number and mean user throughput characterizing each base station and averagedover one particular hour.

• We do an analytic and numerical study for irregular networks, the same one as mentionedin the previous point, considering heterogeneous, multi-tier networks

• We validate our whole approach by comparing our result to these obtained via 3GPP simu-lations [3] and real-field measurements from Orange network and present some advantagesof the developed methods compared to a pure simulation approach.

Most of the time in this chapter we will be interested in the mean user throughput as a keyQoS metric. Mean user throughput is a key quality-of-service metric in cellular data networks.It describes the average “speed” of data transfer during a typical data connection. It is usuallydefined as the ratio of the average number of bits sent (or received) per data request to theaverage duration of the data transfer. Since coexisting connections in a given network cell sharesome given cell transmission capacity, mean user throughput depends inherently on the requesteddata traffic. It also depends on the network architecture (positioning of the base stations) andin fact may significantly vary across different network cells. Moreover, extra-cell interferencemakes performance of different cells interdependent. Predicting the mean user throughput asfunction of the mean traffic demand locally (for each cell) and globally in the network (whichinvolves appropriate spatial averaging in conjunction with the temporal one, already present inthe classical definition of the throughput) is a key engineering task in cellular communications.We will use this metric in a two-fold manner, as an indicator of the network performance and asa dimensioning constraint.

Little’s law allows to calculate the mean user throughput as the ratio of the mean trafficdemand (number of bits requested per unit of time) to the mean number of users in the steadystate of the network. This argument can be used to express mean user throughput locally inany region of the network. Using this argument along with some others presented in the thesis,we show in Section 3.4 how to perform a network dimensioning (planning) assuming all cells”are the same”, i.e. considering a symmetric spatial pattern of base stations. If a network issymmetric, a hexagonal network for example, then all cells are of equal are if we do not considerthe shadowing. On the other hand, shadowing is the same random process for all cells in thenetwork. So, any cell is a statistical representative of the network. Consequently, consideringonly one cell we can capture the network performance in terms of the mean user throughput andstudy the dimensioning of the network. The obtained results are compared to 3GPP simulationsin Section 3.7.2.

Regarding non-symmetric networks, treated in Section 3.5, we introduce the notion of a”typical cell” as a statistical representative of the network. The problem arising here consists in

3.2. RELATED WORK 43

adequate averaging including the spatial one and not only the temporal one. In this case, basestations form some irregular spatial point pattern.

Spatial averages of point patterns, modeling in our case the geographic locations of basestations, can be studied using the formalism of Palm distributions naturally related to the ergodicresults for point processes. Within this setting one considers a typical base station with its typicalcell (zone of service) whose probabilistic characteristics correspond to the aforementioned spatialaverages of the characteristics for all base stations in the network. Adopting this formalism,we define the mean user throughput in the infinite ergodic network as the limit of the ratio ofthe mean volume of the data request to the mean service duration in a large, increasing to thewhole plane, network window. As the main result, we prove that such defined (macroscopic)throughput characteristic is equal to the ratio of the mean traffic demand to the mean numberof users in the typical cell of the network. Both these means account for double averaging: overtime and network geometry.

In Section 3.7.3 we compare the results regarding the QoS in irregular networks to real-field measurements. More precisely, statistics usually collected in operational networks allowto estimate the mean traffic demand and the mean number of users for each cell and hour ofthe day. Even if they carry important information about the local network performance, theyexhibit important variability over time (24 hours) and network cells; cf. Figure 3.8. This can beexplained by the fact that mean user throughput in a particular cell does not depend only on thetraffic in this cell, but also on the neighbouring cells. Moreover, the geometry of different cellsin a real network may significantly differ. For these two reasons, the family of local (establishedfor each cell) throughput-versus-traffic laws usually exhibits a lot of variability both in real dataand in network simulations, and hence does not explain well the macroscopic (network- level)relation between the mean traffic demand and mean user throughput. Finding such a macroscopicrelation in irregular networks is an important task for network dimensioning. It is clear that anappropriate spatial averaging analogous to the one considered in the analytic model is necessaryto discover such a macroscopic law.

A key element of the analysis of the cellular network is the spatial distribution of the signal-to-interference-and-noise ratio (SINR). We show how this distribution enters into the macroscopiccharacterization of the throughput. When considering SINR we are able to account for the factthat the base stations which are idling, i.e., have no users to serve, do not contribute to theinterference. This makes the performance of different cells interdependent and we take it intoaccount via a system of cell-load fixed point equations in Section 3.5.3.

Finally, we show how to amend the model letting it account for the shadowing in the pathloss. The latter is known to impact the geometry of the network, in the sense that the servingbase station is not necessarily the closest one. It also alters the distribution of the SINR.

3.2 Related work

3.2.1 Related work regarding the dimensioning problem

The dimensioning of cellular networks is often treated using a coverage or static capacity ap-proach. Basically one aims to assure that the bit-rate (or the SINR) of a permanent user exceedssome target value with a high probability. To do so, in [52] the (CDF) of the so-called effectiveSINR (‘averaged’ over the different OFDM subcarriers) is calculated with the help of a Gaussianapproximation. Then this CDF is used to assure the coverage condition. A similar approach isadopted in [66] where other approximations for the CDF of the SINR are proposed. In theseworks arrivals and departures of users are not considered. This dynamic context is taken intoaccount in the present chapter.


The dimensioning problem in this dynamic context may, in principle, be solved using a simu-lation approach such as that proposed in [3]. There are some other simulation tools (not neces-sarily compliant with 3GPP) such as LTE-Sim [77] developed by TelematicsLab, LTE simulatordeveloped by University of Vien [72,86] and LENA tool [12,13] developed by CTTC.

However the simulation approach requires a huge amount of time and it is useless in thecontext of dimensioning. Indeed, calculating the users quality of service for a particular networkconfiguration by 3GPP simulations takes up to a few weeks of calculation, and thus the dimen-sioning problem, which requires tens of such calculations, would take about a year! Analyticalternatives to pure simulations have already been proposed for VBR calls. They are essentiallybased on queueing theory.

The dynamics of user arrivals and departures are taken into account in [25], [54], [58] assumingthe symmetric network pattern and that base stations are always transmitting at their maximalpower and . In this context, the peak bit-rate at a given location is defined as the bit-rateobtained by a user assumed alone in the cell. The quality of service perceived by the usersin the long run of their arrivals and departures is then calculated using multi-class processorsharing models [33], [26, Proposition 3.1]. The effect of mobility on the users’ QoS is studiedin [24, §4], [60].

In reality, the base stations emit only when they have at least one user to serve, and thusinterference depends on the traffic of other base stations. In order to account for this dependence,the authors of [82] describe a fixed-point problem and propose to solve it iteratively.

3.2.2 Related work regarding QoS evaluation

The evaluation of user QoS metrics in cellular networks is a hard problem, but crucial for networkoperators and equipment manufacturers. It also motivates a lot of engineering and researchstudies and as dimensioning can be done using similar types of simulations as mentioned above.

A possible analytical approach to this problem is based on the information theoretic char-acterization of the individual link performance; cf e.g. [53, 73], in conjunction with a queueingtheoretic modeling and analysis of the user traffic; cf. e.g. [24–26, 54, 62, 82]. All these worksconsider some particular aspects of the network and none of them considers a large, irregularmulti-cell network. Such a scenario is studied in our approach by using stochastic-geometrictools combined with the two aforementioned theories. As a result, we propose a global, macro-scopic approach to the evaluation of the user QoS metrics in cellular networks, which we compareand validate with respect to real network measurements. Stochastic geometry has already beenshown to give analytically tractable models of cellular networks, see e.g. [8].

The disparity of cell load and QoS parameters has already been observed in the literature. Forexample [97] shows temporal and spatial cell load fluctuations in cellular commercial networks.These results are obtained from data collected by the mobile operators. In [76], traffic and cellload disparity are shown graphically. Data are derived from nationwide 3G cellular network andthe results are presented from network and user point of view. In [48] the authors analyzedQoS (throughput etc.) perceived by the users using data collected from mobile operators andexperiments. QoS parameters such as throughput, latency etc. are also analyzed based onfield-measurements in [89]. Cell load and QoS parameters disparity are assumed in many studiestreating load balancing. Load balancing consists in the redistribution of load between cells in sucha way that all cells are equally loaded. Namely, articles as [74], [41], [96] and [103] present differentalgorithms for spectrum and energy efficient load balancing. The performance of heterogeneousnetworks gained a lot of research interest recently, for example see [31] and [50], since theirdeployment is already commercial and will probably continue to grow. In [70] the authors givean algorithm for network planning implying cell load disparity such that to compensate spatially

3.3. PROCESSOR-SHARING QUEUE MODEL FOR ONE CELL SCENARIO 45

non-uniform traffic demand, but they do not give the CDF of cell load and other QoS parameters.The authors of [62] and [87] describe the dependence between the traffic demand and the

interference in wireless cellular networks and show that there is a fixed-point problem in theexpressions giving the cell load.

3.3 Processor-sharing queue model for one cell scenario

We consider a cell comprising a finite set 1, 2, . . . , J of possible locations. We denote by Rj thepeak bit-rate at each location j ∈ 1, 2, . . . , J of the cell; that is the bit-rate allocated by thebase station to the user at this location assuming that: (1) the user is alone in the consideredcell; and (2) the other base stations transmit at their maximal powers (this assumption will berevisited later). The peak bit-rate can be e.g. the outcome of the link analysis in Chapter 2.

We describe now the allocation of the resources to the different users present in the cell at agiven time. Let xj be the number of users at location j and x = (x1, x2, . . . , xJ) be the vectorcounting the number of users at each location called configuration of the users. Assume that thebase station allocates to each user at location j a specific portion of time ϕj depending on itslocation, then such user gets the bit-rate

rj = ϕjRj (3.1)

Writing that the sum of the time portions may not exceed 1; i.e.∑Jj=1 xjϕj ≤ 1, we get the

following constraint on the bit-rates which may be allocated by the base station to the differentusers in its cell

J∑j=1

xjrjRj≤ 1 (3.2)

We shall assume that each user gets an equal portion of time ϕj = 1/N where N is the totalnumber of users in the cell; then we deduce from Equation (3.1)

rj =RjN, j ∈ 1, 2, . . . , J (3.3)

Remark 7 The constraint (3.2) (and the particular allocation (3.3)) may also be obtained bymultiplexing the users in frequency or codes (or any mixture of time, frequency and code multi-plexing). The only condition is that the users are served in a strictly orthogonal way. Moreover,the bit-rates rj should be understood as an average over a sufficiently long run of the multiplexing.

We now introduce the dynamics of user (call) arrivals and departures. The inter-arrival timesat location j are assumed to be exponentially distributed random variables with parameter λj(average inter-arrival duration equals 1/λj). The users arriving to location j require to transmitsome volumes of data (in bits) which are i.i.d. random variables of mean 1/µj , not necessarilyexponentially distributed. We assume independence between the inter-arrivals of users and therequired volumes. We call ρj := λj/µj the traffic demand at location j and ρ =

∑Jj=1 ρj the

total traffic demand in the cell. We denote by λ =∑Jj=1 λj the total arrival rate.

3.3.1 No mobility case

We assume in the present section that the users do not move during their calls. The followingproposition gives the performance in the long run of the calls arrivals and departures. Denotethe set of locations by D := 1, 2, . . . , J. In order to position our problem in the queueing


theory context, we may view a cell as a queue and a location as a class. In doing so, a cell maybe considered as a multi-class processor sharing queue. We define a critical traffic demand asfollows:

ρc :=ρ∑J

j=1 ρjR−1j

(3.4)

Proposition 4 The cell is stable if and only if the traffic demand does not exceed the criticaltraffic demand; that is

ρ

ρc=

J∑j=1

ρjR−1j < 1 (3.5)

In case of stability, the steady state distribution of the configuration of the users is

π (x) =

(1− ρ

ρc

)xD!∏j∈D

(ρj/Rj)xj

xj !, x ∈ ND (3.6)

where x = (xj)j∈D is a vector counting the numbers of users in each location and xD :=∑j∈D xj.

Moreover, the mean number of users, the delay and the throughput per user at a given locationj ∈ 1, 2, . . . , J are respectively given by

Nj =ρj(

1− ρρc

)Rj, Tj =

1(1− ρ

ρc

)Rjµj

, rj =

(1− ρ

ρc

)Rj (3.7)

and the mean number of users, the delay and the throughput per user in the cell in the steadystate are respectively given by

N =ρ

ρc − ρ, T =

ρ

(ρc − ρ)λ, r = ρc − ρ (3.8)

Remark 8 It might look cumbersome to express the stability condition (3.5) in terms of ρρc

.This is particularly convenient when the traffic demand ρj at location j is parameterized in thefollowing manner:

ρj = ρdj , j ∈ 1, 2, . . . , J (3.9)

where dj is a geographical distribution of traffic and ρ is a parameter expressing the total trafficdemand. In this case

ρc :=

J∑j=1

djR−1j

−1

(3.10)

and the condition 3.5 can be written as:ρ < ρc (3.11)

In the remaining part we will always assume traffic demand in the form of (3.9).

Proof of Proposition 4. See the appendix 3.A for a detailed proof in the Markovian case;i.e., when the transmitted volumes are assumed exponentially distributed. In the more generalcase (when the transmitted volumes are arbitrary distributed) the proof is more involved. Forthe stability condition (3.11) and the expression (3.6) of the steady state distribution see [33], [26,Proposition 3.1].

The mean number of users, the delay and the throughput expressions may be obtainedfrom [25] or by specializing [58, Example 10] to the current discrete context with no mobility.

3.3. PROCESSOR-SHARING MODEL 47

We give here an outline of the proof of Equations (3.7) and (3.8). The mean number of users(either in a given location or in the cell) is obtained from the expression (3.6) of the steady statedistribution. The delays are then deduced from Little’s formula [11]

Tj =Njλj, T =

N

λ

The throughput per user at a given location j is simply the average volume 1/µj divided by thedelay Tj . It remains to show the expression of the throughput per user in the cell; i.e. r = ρc−ρ.To do so, observe that the throughput of the whole cell at the steady state is equal to the trafficdemand ρ; since at equilibrium the volumes of data incoming to and leaving the cell in the longrun should be equal. The throughput per user in the cell is defined as the ratio of the cellthroughput ρ by the average number of users; that is r = ρ

N= ρc − ρ.

Note that (3.6) may be written as follows

π (x) = [(1− ρ′) ρ′xD ]

xD!

J∏j=1

(ρ′j/ρ

′)xjxj !

, x ∈ NJ

where ρ′j = ρj/Rj and ρ′ = ρ/ρc. If follows that the distribution of the total number of users

in the cell XD :=∑Jj=1Xj is the geometric distribution on N with parameter 1 − ρ′ = 1 − ρ

ρc;

that is Pr (XD = n) = (1− ρ′) ρ′n, n ∈ N. In particular the probability that the cell is not emptyequals ρ′ = ρ

ρc(called load of the cell).

Moreover the above expression shows that, given the total number of users n, the distributionof the number of users among the different locations is multinomial of size (n, J) and parameters(ρ′1/ρ

′, . . . , ρ′J/ρ′); this is equivalent to say that the users are assigned to classes independently

of each other, with the probability ρ′j/ρ′ of a given user to be assigned to class j.

Corollary 3 With the notations of Proposition 4, if ρ < ρc then

r =ρ∑J

j=1 ρj r−1j

and

T =1

λ

J∑j=1

λj Tj

where λ =∑Jj=1 λj is the total arrival rate to the cell.

Proof. Straightforward calculations from (3.7) and (3.8).The above corollary shows that the throughput per user in the cell is the harmonic mean of

the throughputs at the different locations weighted by the traffic demands; whereas the delayper user in the cell is the arithmetic mean of the delays at the different locations pondered bythe arrival rates. So we should be careful when calculating the average of the quality of serviceover a cell.

Mobile categories

A user located at a given geographic location undergoes some radio conditions; i.e., some specificpropagation losses (due to distance, shadowing and indoor) with the different base stations inthe network. Given these radio conditions, the user gets some bit-rate. The relation between the


radio conditions and the bit-rate may be specific to each mobile category. We shall use the termclass to designate not only the geographic location but also the specific mobile’s category.

Let K be the number of mobile categories and I be the number of geographic locations, thena class j is a pair (i, κ) where i ∈ 1, 2, . . . , I is a geographic location and κ ∈ 1, 2, . . . ,K isthe mobile category. The number of classes is now J = I × K. With this extended notion ofclass, the results of Proposition 4 obviously apply; we get in particular the expression for thethroughput and delay for each class j = (i, κ). The following proposition gives the expressionsfor the throughput and delay per mobile’s category but averaged over the geographic locations.

Proposition 5 Assume the stability condition (3.11). For a given mobile’s category κ ∈ 1, 2, . . . ,K,the throughput per user in the cell in the steady state is

rκ =

∑Ii=1 ρi,κ∑I

i=1 ρi,κr−1i,κ

that is the harmonic mean of the throughputs at the different geographic locations weighted bythe corresponding traffic demands. The delay per user in the cell in the steady state is

Tκ =

∑Ii=1 λi,κTi,κ∑Ii=1 λi,κ

that is the arithmetic mean of the delays at the different geographic locations pondered by thecorresponding arrival rates.

Proof. The result is obtained by specializing [58, Example 10] to the current discrete contextwith no mobility.

Connection between traffic demand, QoS, capacity and cell radius

Fixing a target value r for the throughput per user in the cell, we deduce from (3.8)

ρc − ρ = r

Since ρ and ρc are functions of the cell radius, the above equation might be solved with respectto the cell radius. This approach will be further developed in Section 3.4 and illustrated bynumerical examples in Section 3.7.2.

We assume, without loss of generality, that the peak bit-rates are sorted in decreasing order;that is R1 > R2 > . . . > RJ . Fixing a target value rJ of the throughput per user in the cellborder, we deduce from (3.7)

ρ

ρc= 1− rJ

RJ

which may be taken as the dimensioning constraint.Given some q ∈ [0, 1], let jq be the q-quantile of the traffic distribution (d1, d2, . . . , dJ); i.e.

such thatjq−1∑j=1

dj < q ≤jq∑j=1

dj

Then fixing a target value rjq for the throughput per user at location jq, we deduce from (3.7)

ρ

ρc= 1−

rjqRjq

which may be taken also as the dimensioning constraint.

3.3. PROCESSOR-SHARING MODEL 49

Remark 9 Note that jq is not the q-quantile of the proportion of users in the steady state(N1/N, N2/N, . . . , NJ/N

)since from (3.7)

NjN

=ρc

ρ

ρjRj

= djρc

Rj

Thus we should not say that a proportion q of the users in the steady state would have a throughputlarger than rjq ; but we should say that a proportion q of the cell area (weighted by the trafficdemand) would have a throughput larger than rjq .

3.3.2 Infinite mobility

Now we will study the effect of mobility on performance from the queueing theory point of view.The case when the average user’s speed is finite and nonnull is intractable analytically. But itmay be bounded by the two extreme cases of no mobility and infinite mobility since mobilityimproves performance as proved in [24, §4.2.2]. This motivates our study of the infinite mobilitycase where each user is assumed to move along all the possible locations and thus experiences allthe radio conditions during his call (whereas in the no mobility case, the user undergoes a givenradio condition).

We assume in the present section that the mean volume of data does not depend on thelocation; that is µj ≡ µ. We assume also that each user moves according to some ergodicMarkov process with invariant distribution (%1, %2, . . . , %J). Moreover we assume that each user

moves so fast that he receives a peak bit-rate averaged over his mobility; that is∑Jj=1 %jRj .

Then the bit-rate allocation (3.3) is now replaced by

rj ≡ r :=

∑Jj=1 %jRj

N, j ∈ 1, 2, . . . , J (3.12)

Proposition 6 In case of infinite mobility, the cell is stable when

ρ < ρc

where

ρc :=

J∑j=1

%jRj (3.13)

In case of stability, the mean number of users, the delay and the throughput per user in the cellat the steady state are respectively given by

N =ρ

ρc − ρ, T =

ρ

(ρc − ρ)λ, r = ρc − ρ

Proof. See [60, Proposition 2].

Note that ρc is the arithmetic mean of the peak bit-rates weighted by the mobility distribution(%1, %2, . . . , %J). Assume that the traffic demand (ρ1, ρ2, . . . , ρJ) is proportional to the mobilitydistribution, then, since the arithmetic mean is larger than the harmonic mean, we deduce thatthe critical traffic with mobility is larger than the critical traffic in the no mobility case which isconsistent with [24, §4.2.2].


Mobile categories

If there are different mobile categories, then it is natural to assume that mobility holds betweenthe geographic locations 1, 2, . . . , I but not between categories 1, 2, . . . ,K. We assume thatthe mean volume of data does not depend on the geographic location but may depend on themobile’s category; that is

µi,κ ≡ µκ, i ∈ 1, 2, . . . , I , κ ∈ 1, 2, . . . ,K

Again we assume that each user of category κ ∈ 1, 2, . . . ,K moves so fast that he receives apeak bit-rate averaged over his mobility; that is

Rκ :=

I∑l=1

%l,κRl,κ, κ ∈ 1, 2, . . . ,K (3.14)

Thus the bit-rate allocation is now

ri,κ ≡ rκ :=

∑Il=1 %l,κRl,κ

N, i ∈ 1, 2, . . . , I , κ ∈ 1, 2, . . . ,K

Proposition 7 The cell is stable when

ρ < ρc :=ρ∑K

κ=1 ρκR−1κ

where Rκ are given by (3.14) and

ρκ :=

I∑l=1

ρl,κ, κ ∈ 1, 2, . . . ,K

is the cell traffic for category κ. In case of stability, the mean number of users, the delay and thethroughput per user of category κ ∈ 1, 2, . . . ,K are respectively given by

Nκ =ρκ(

1− ρρc

)Rκ

, Tκ =1(

1− ρρc

)Rκµκ

, rκ =

(1− ρ

ρc

)Rκ

and the mean number of users, the delay and the throughput per user in the cell in the steadystate are respectively given by

N =ρ

ρc − ρ, T =

ρ

(ρc − ρ)λ, r = ρc − ρ

Proof. Observe that the present context is similar to that of Proposition 4 with the categorieshere in the role of the locations there and where the peak bit-rates are given now by (3.14). Thedesired results then follow from Proposition 4.

3.4 Multi-cell scenario: symmetric networks

An LTE cellular network is composed of base stations covering some geographic zone. Each basestation transmits at some power limited to some maximal value P and assigns a specific portionw of the total system bandwidth W to each user. Here, we consider that all cells have the sameform.

3.4. MULTI-CELL SCENARIO: SYMMETRIC NETWORKS 51

Since fading is already averaged out at the link level, the remaining propagation loss comprisesonly the distance and shadowing. Consider a given user and let Lv be his propagation loss withbase station v. We assume that each user is served by the base station (denoted by index u)with the smallest loss; that is Lu = inf Lv. Assume moreover that each base station transmitsa constant power spectral density.

We assume in the present section that each base station transmits at its maximal power PThen the received signal power equals

P =w

W

P

Lu

and the interference equal

I =w

W

∑v 6=u

P

Lv

Let N be the noise power in the system bandwidth, then the SINR per transmitting antenna1

equals

SINR =PtA

wWN + I

tA

=1

NLutAP

+ f(3.15)

where

f :=∑v 6=u

LuLv

(3.16)

is called the interference factor. The SINR calculated by Equation (3.15) should be plugged inEquation (2.19) to get the corresponding bit-rate.

3.4.1 Cell load

In this Section we will explain the dependence between cells in a given LTE network, and proposea fixed-point equation which can capture this dependance. We assumed in the previous sectionthat the interfering base stations transmit always at their maximal power P . In fact a basestation does not transmit when there are no users to serve. The power transmitted by basestation v is then 1 Xv (t) 6= 0P where Xv (t) is the number of users served by base station vat time t.

Thus the interference at time t equals

I (t) =w

W

∑v 6=u

1 Xv (t) 6= 0 PLv

Explicit analysis of the multi-cell model with assumption is not possible. Even the stabilitycondition of the network in this case is not yet known. Nevertheless, the full activity assumptionmade in the previous section gives a useful lower bound for the peak bit-rates and thus a lowerbound of the critical traffic demand. We shall make now another model assumption.

1See [94, Equation (3.169)].


Invoking the law of large numbers, we may approximate the interference as follows

I (t) ' w

W

∑v 6=u

E [1 Xv (t) 6= 0] PLv

=w

W

∑v 6=u

Pr (Xv (t) 6= 0)P

Lv

=w

W

∑v 6=u

ρ

ρc

P

Lv

where for the third equality we use the observation following Proposition 4. Then the SINRequals

SINR =1

NL0

tAP+ ρ

ρcf

and the corresponding peak bit-rate equals

R = Wψ(SINR) = Wψ

(1

NL0

tAP+ ρ

ρcf

)(3.17)

where the function ψ is given for example by (2.17).Equations (3.4) and (3.13) show that the critical traffic ρc is a function of the peak bit-rates

which are themselves functions of the critical traffic as shown in the above equation. Thus ρc

may be obtained by solving a fixed-point problem. For example, in the case of infinite mobilityEquation (3.13) implies

ρc = E [R] = E

[Wψ

(1

NL0

tAP+ ρ

ρcf

)](3.18)

where the expectation is with respect to a user distributed according to the mobility invariantdistribution %. Once the above fixed-point problem is solved, the ratio

θ :=ρ

ρc(3.19)

is called the load of the system. The equation given by (3.18) we call fixed-point equation.

Definition 1 The following different load situations are considered in conjunction with thequeueing approach:

• Adapted (Weighted) interference (load): A base station transmits only when it has at leastone user to serve.

• Full interference (load): Base stations are always transmitting at their maximal power.

• Null interference (load): Interference is assumed completely cancelled. This corresponds toa cell in isolation.

Remark 10 Note that the load depends on the traffic demand, so we can not consider these twoparameters as independent inputs when evaluating the users QoS.

Remark 11 The above queueing analysis is carried out for a typical cell of a network composedof multiple cells assumed statistically equivalent. Indeed, the interference between the differentcells is taken into account through the interference factor (3.16) and the solution of the fixed-pointproblem (3.18).

3.5. MULTI-CELL IRREGULAR NETWORKS SCENARIO 53

3.5 Multi-cell irregular networks scenario

In contrast to Section 3.4, here we consider irregular cellular networks, which corresponds betterto the commercially deployed networks, especially in urban areas. Irregular means here that aspatial pattern of base stations’ locations is non-symmetric and that different base stations radiatedifferent powers (in the first part of the analysis we will consider constant radiation power).

3.5.1 Network model

We consider locations v1, v2, . . . of base stations (BS) on the plane R2 as a realisation of a pointprocess, which we denote by Φ. 2 We assume that Φ is stationary and ergodic with positive,finite intensity (mean number of BS per unit of area) γ. 3

In order to simplify the presentation, we shall make first the following two assumptions, whichwill be relaxed in Sections 3.5.3 and 3.5.3, respectively.

1. There is no shadowing. The (time-averaged over fading) propagation loss depends only onthe distance r between the transmitter and the receiver through a path-loss function l(r),which we assume increasing. The SINR expression remains valid except that instead ofpropagation loss, we consider only the path-loss.

2. Full load (interference). Each base station is always transmitting at some fixed power P ,common for all stations.

We will also assume throughout the whole Section that each user is served by the BS which heor she receives with the strongest signal power. The consequence of the assumption 1 above is thateach BS u ∈ Φ serves users in a geographic zone V (u) = y ∈ R2 : |y − u| ≤ minv∈Φ,v 6=u |y − v|which is called Voronoi cell of u in Φ.

For single link performance, we use formula (2.17) from Section 2.5. We assume the sameservice policy and traffic demand as in Section 3.3. We assume spatially uniform traffic demand,which means that different base stations have different traffic demands to serve. The trafficdemand in a given cell equals

ρ (v) = ρ |V (v)| , v ∈ Φ. (3.20)

3.5.2 Generalization of processor sharing model

In this section, we generalize the results already presented in Section 3.3. Namely, there weconsider the case when the cell is stable, that is (3.11) is always valid. Here, we consider also thepossibility that there are unstable cells present in a network. Another difference is that evidentlyhere different base stations serve zones (cells) of different size even without considering the shad-owing. For a fixed configuration of BS Φ, the service of users arriving to the cell V (v) of a givenBS v ∈ Φ can be modeled by an appropriate (spatial) multi-class processor sharing queue, withclasses corresponding to different peak bit-rates characterized by user locations y ∈ V (v). Notealso that a consequence of our model assumptions (in particular the full interference assump-tion 2, inter-cell channel independence and space-time Poisson arrivals) the service processes ofdifferent queues are independent.

2According to the formalism of the theory of point processes (cf e.g. [37]), a point process is a random measureΦ =

∑j δXj , where δx denotes the Dirac measure at x.

3Stationarity means that the distribution of the process is translation invariant, while ergodicity allows tointerpret some mathematical expectations as spatial averages of some network characteristics.


We consider now the steady-state number of users served in each cell V (v). 4 The followingexpressions follow from the queueing-theoretic analysis of the processor sharing systems of eachBS v ∈ Φ, explained in Section 3.3.1, also, cf [25, 62] for the details.

• The service process of BS v ∈ Φ is stable if and only if its traffic demand does not exceedthe critical value that is the harmonic mean of the peak bit-rate over the cell:

ρc (v) :=|V (v)|∫

V (v)1/R (SINR (y,Φ)) dy

. (3.21)

Note that ρc(v) depends on v and on Φ. The same observation is valid for the subsequentcell characteristics. This definition comes from (3.10).

• The mean user throughput in the given cell, defined as the ratio of the mean volume ofthe data request 1/µ to the average service time of users in this cell, can be expressed asfollows

r (v) = max(ρc (v)− ρ (v) , 0) . (3.22)

• The mean number of users in steady state of the given cell equals to

N (v) =ρ (v)

r (v). (3.23)

Note that N(v) =∞ if ρ(v) ≥ ρc(v).

• The probability that the given BS is not idling in steady state (has at least one user toserve) equals

p (v) = min (θ (v) , 1) , (3.24)

where θ(v), which we call cell load, is defined as

θ (v) :=ρ (v)

ρc (v). (3.25)

Note that the cell is stable if and only if θ(v) < 1 and

θ(v) = ρ

∫V (v)

1/R (SINR (y,Φ)) dy . (3.26)

Moreover,

N(v) =θ(v)

1− θ(v), (3.27)

r(v) = ρ(v)(1/θ(v)− 1) (3.28)

provided θ(v) < 1. The function R is the same as in (3.17).The above expressions allow to express all other characteristics in terms of the traffic demand

per cell ρ(v) and the cell load θ(v).

Remark 12 All the above characteristics are local network characteristics in the sense that theycharacterize the service at each BS v and vary over v ∈ Φ. Real data analysis and simulationsfor Poisson network models exhibit a lot of variability among these characteristics. In particular,plotting the mean user throughput r(v) as function of the mean traffic demand ρ(v) for differentv ∈ Φ does not reveal any apparent systematic relation between these two local characteristics;cf. Figure 3.8.

4Note that the (mean) QoS characteristics of users in this state correspond to time-averages of user character-istics.


3.5.3 Global network characteristics

In this section we propose some global characteristics of the network allowing to characterize itsmacroscopic performance. We are particularly interested in finding such a relation between the(per area) traffic demand ρ and the (global) mean user throughput in the network, with thislatter characteristic yet to be properly defined.

Typical cell of the network

A first, natural idea in this regard is to consider spatial averages of the local characteristics inan increasing network window A, say a ball centered at the origin and the radius increasing toinfinity. Assuming ergodicity of the point process Φ of the BS, these averages can be expressedand calculated as Palm-expectations of the respective characteristics of the so called “typicalcell” V (0). For example

lim|A|→∞

1/Φ(A)∑v∈A

ρ(v) = E0[ρ(0)] = ρE0[|V (0)|] . (3.29)

The typical cell V (0) is the cell of the BS located at the origin v = 0 and being part of thenetwork Φ distributed according to the Palm distribution Pr0 associated to the original stationarydistribution Pr of Φ. In the case of Poisson process, the relation between the Palm and stationarydistribution is particularly simple and (according to Slivnyak’s theorem) consists just in addingthe point v = 0 to the stationary pattern Φ.

The convergence analogue to (3.29) holds for each of the previously considered local char-acteristics E0[ρc(0)], E0[r(0)], E0[N(0)], E0[p(0)] and E0[θ(0)]. The convergence is Pr almostsure and follows from the ergodic theorem for point processes (see [9, Theorem 4.2.1], [37, The-orem 13.4.III]). However, as we will explain in what follows, not all of these mean-typical cellcharacteristics have natural interpretations as macroscopic network characteristics.

First, note that the existence of some (even arbitrarily small) fraction of BS v which are notstable (with ρ(v) ≥ ρc(v), hence N(v) =∞) makes E0[N(0)] =∞.

Remark 13 For a well dimensioned network one does not expect unstable cells. For a perfectlyhexagonal network model Φ all cells are stable or unstable depending on the value of the per-area traffic demand ρ. An artifact of an infinite, homogeneous, Poisson model Φ is that forarbitrarily small ρ there exists a non-zero fraction of BS v ∈ Φ, which are non-stable. Thisfraction is very small for reasonable ρ, allowing to use Poisson model to study QoS metricswhich, unlike E0[N(0)], are not “sensitive” to this artifact.

We will also show in the next section that it is not natural to interpret E0[r(0)] (which is notsensitive to the existence of a small fraction of unstable cells) as the mean user throughput inthe network; see Remark 16. Before we give an alternative definition of this latter QoS, let usstate the following result, which will be useful in what follows.

Proposition 8 We have

E0[ρ(0)] =ρ

γ, (3.30)

E0[θ(0)] =ρ

γE[1/R (SINR (0,Φ))] . (3.31)

Proof. The first equation is quite intuitive: the average cell area is equal to the inverse of theaverage number of BS per unit of area. Formally, both equations follow from the inverse formulaof Palm calculus [9, Theorem 4.2.1]. In particular, for (3.31) one uses representation (3.26) inconjunction with the inverse formula.


Remark 14 Note that the expectation in the right-hand-side of (3.31) is taken with respect to thestationary distribution of the BS process Φ. It corresponds to the spatial average of the inverse ofthe peak bit-rate calculated throughout the network. The (only) random variable in this expressionis the SINR experienced by the typical user. This distribution is usually known in operationalnetworks (estimated from user measurements). It can also be well approximated using a Poissonnetwork model for which its distribution function admits an explicit expression [19].

Mean user throughput in the network

Faithful to the usual definition of the mean user throughput as the ratio of the mean volumeof the data request to the mean service duration (which we retained at the local, cell level) weaim to define now the mean user throughput in the (whole) network as the ratio of these twoquantities taken for increasing network window A. However in order to “filter out” the impactof cells which are not stable and avoid undesired degeneration of this characteristic (e.g. forPoisson process; cf. Remark 13) let us consider the union of all stable cells

S :=⋃

v∈Φ:ρ(v)<ρc(v)

V (v) .

Note that the stationarity of Φ implies the same for the random set S. We denote by πS =E[1(0 ∈ S)] the volume fraction of S and call it the stable fraction of the network. It is equalto the average fraction of the plane covered by the stable cells; cf. [9, Definition 3.4 and thesubsequent Remark]. Denote also

N0 := E0[N(0)1(N(0) <∞)] .

We are ready now to define the mean user throughput in the network r0 as the ratio of the averagenumber of bits per data request to the average duration of the data transfer in the stable partof the network

r0 := lim|A|→∞

1/µ

(temporal-)mean service time in A ∩ S. (3.32)

Here is the key result of the typical cell approach. Its proof is given at the end of this section.

Theorem 1 For an ergodic network Φ we have

r0 =ρ πSγN0

. (3.33)

Remark 15 Equation (3.33) provides a macroscopic relation between the traffic demand and themean user throughput in the network, which we are primarily looking for. It will be validated bycomparison to real data measurements. The quantities N0 and πS do not have explicit analyticexpressions analogous to (3.31). Nevertheless they can be estimated from simulations of a givennetwork model Φ. Note that these are static simulations of the network model. No simulation ofthe traffic demand process is necessary, which greatly simplifies the task. For small and moderatevalues of the traffic demand (observed in real networks) one obtains πS ' 1. Moreover, inSection 3.5.4 we will propose some more explicit approximation of N0.

Remark 16 Assume that there are no unstable cells in the network. This is the case e.g. forlattice (say hexagonal) network models with traffic demand ρ < ρc(v) = ρc, where the value ofthe critical traffic is the same for all cells. Then πS = 1, N0 = E0[N(0)] and the relation (3.33)takes form

r0 =ρ

γE0[N(0)]=

E0[ρ(0)]

E0[N(0)]. (3.34)


Thus, in general r0 6= E0[r(0)] = E0[ρ(0)/N(0)]. We want to emphasize that this is not merelya theoretical detail resulting from our (and common) definition of the mean throughput (3.32).The expression E0[r(0)] = E0[ρ(0)/N(0)], which in principle can be considered as another globalQoS metric, is in practice difficult to estimate. Indeed, when estimating E0[r(0)] as the averageof the ratio “traffic demand to the number of users” from real data measurements, one needs togive a special treatment to observations which correspond to cells during their idling hours (i.e.,with no user, and such observations are not rare in operational networks). Neither skipping norliteral acceptance of these observations captures the right dependence of the mean user throughputon the traffic demand.

Proof of Theorem 1. By Little’s law [11] the temporal mean service time TW of usersin any region of the network W, say the union of stable cells with BS in some region A, W =⋃v∈A∩S V (v), is related to the mean number NW of the users served in this region W in the

steady state by the equation NW = λ|W|TW . Consequently, the mean user throughput in thisregion W can be expressed as 1/(µTW) = ρ|W|/NW . Using

|W|NW

=|W|∑

v∈A∩S N(v)

=

∑v∈A |V (v)|1(N(v) <∞)

|A||A|∑

v∈A∩S N(v)

and again the ergodic theorem for point yprocess Φ, we obtain that the limit in (3.32) is Pr-almost surely equal to ρE0[|V (0)|1(N(0) <∞)]/E0[N(0)1(N(0) <∞)]. By the aforementionedinverse formula of Palm calculus we conclude E0[|V (0)|1(N(0) <∞)] = E[1(0 ∈ S)]/γ.

Cell-load equations

We have to revoke now the full interference assumption 2 made in Section 3.5.1. An amend-ment is necessary in this matter for the model to be able to predict the real network data; cfnumerical examples in Section 3.7. Recall that the consequence of this assumption is that in theexpression (3.15) of the SINR all the interfering BS are always transmitting at a given powerP . In real networks BS transmit only when they serve at least one user. 5 Taking this factinto account in an exact way requires introducing in the denominator of (3.15) the indicatorsthat a given station v ∈ Φ at a given time is not idling. This, in consequence, would lead tothe probabilistic dependence of the service process at different cells and result in a non-tractablemodel. In particular, we are not aware of any result regarding the stability of such a family ofdependent queues. For this reason, we take into account whether v is idling or not in a simplerway, multiplying its powers P by the probability p(v) that it is not idle in the steady state. Inother words we modify the expression of the SINR as follows (for base station u as a serving basestation)

SINR (y,Φ) :=P/l (|y − u|)

N + P∑

v∈Φ\up (v) /l (|y − v|)

, (3.35)

for y ∈ V (v), v ∈ Φ where p(v) are cell non-idling probabilities given by (3.24). We will seein Section 3.7 that this model, called weighted interference (load) model, fits better to real fieldmeasurements than the full interference model. The above modification of the model preservesthe independence of the processor-sharing queues at different cells given the realization Φ of thenetwork (thus allowing for the explicit analysis of Section 3.5.2). However the cell loads θ(v) are

5Analysis of more sophisticated power control schemes is beyond the scope of this thesis.


no longer functions of the traffic demand and the SINR experienced in the respective cells, butare related to each other by the following equations that replace (3.26)

θ (v) = ρ

∫V (v)

1

R

(P/l(|y−u|)

N+P∑

v∈Φ\umin(θ(v),1)/l(|y−v|)

) dy . (3.36)

We call this system of equations in the unknown cell loads θ (v)v∈Φ the cell-load equations.

Remark 17 (Spatial stability) The weighted interference model introduces more “spatial” de-pendence between the processor sharing queues of different cells, while preserving their “temporal”(conditionally, given Φ) independence. A natural question regarding the existence and uniquenessof the solution of the fixed point problem (3.36) arises. Note that the mapping in the right-hand-side of (3.36) is increasing in all θ(v), v ∈ Φ provided function R is increasing. Using thisproperty it is easy to see that successive iterations of this mapping started off θ(v) ≡ 0 on onehand side and off θ(v) as in (3.26) (full interference model) on the other side, converge to aminimal and maximal solution of (3.36), respectively. An interesting theoretical question regardsthe uniqueness of the solution of (3.36), in particular for a random, say Poisson, point process Φ.Answering this question, which we call “spatial stability” of the model, is unfortunately beyondthe scope of this thesis.6 The simulation study of the typical cell model, presented in Section 3.7(where we use Matlab to find a solution of (3.36) for any given finite pattern of base stations Φ)is less stable for larger values of the traffic demand ρ.

In the mean cell approach (cf Section 3.5.4) we take into account the weighted interferencemodel by the following (single) equation in the mean-cell load θ

θ =ρ

γE

[1/R

(P/l (|v∗|)

N + P∑v∈Φ\v∗ θ/l (|y − v|)

)](3.37)

where v∗ is the location of the BS whose cell covers the origin. We solve the above equation withθ as unknown. We will show in the numerical section that the solution of this equation gives agood estimate of the empirical average of the loads θ (v)v∈Φ obtained by solving the system ofcell-load equations (3.36) for the simulated model.

Remark 18 (Pilot channel) The cells which are not idle might still emit some power (e.g. inthe pilot channel). This can be taken into account by replacing p(v) = min(θ(v), 1) in (3.36) byp(v)(1 − ε) + ε, where ε is the fraction of the power emitted all the time. Similar modificationconcerns θ in the right-hand-side of (3.37).

Shadowing

Until now we were assuming that the propagation loss is only induced by the distance between thetransmitter and the receiver. In this section we will briefly explain how the effect of shadowingcan be taken into account.

Assume that the shadowing between a given station v ∈ Φ and all locations y ∈ R2 ismodeled by some random field Sv (y − v). That is, we assume the propagation loss between v

and y Lv (y) = l(|y−v|)Sv(y−v) . We assume that, given Φ, the random fields Sv(·) are independent

6Existence and uniqueness of the solution of a very similar problem (with finite number of stations and adiscrete traffic demand) is proved in [87].


across v ∈ Φ and identically distributed. In general, we do not need to assume any particulardistribution for Sv(·) (neither independence nor the same distribution of Sv(y) across y).

The assumption that each user is served by the BS received with the smallest path-loss resultsin the following modification of the geographic service zone of u, which we keep calling “cell”

V (u) = y ∈ R2 : Lu (y) ≤ minv∈Φ,v 6=u

Lv (y) . (3.38)

For mathematical consistency we shall assume that, almost surely, the origin belongs to a uniquecell (i.e., is not located on any cell boundary).

The SINR at location y can be expressed by (3.15) or (3.35) with l(|y− v|) replaced by Lv(y)for v ∈ Φ, depending on whether we consider the full interference or the weighted interferencemodel, for y ∈ V (v), with V (v) defined by (3.38). The same modification regards the cell-loadequations (3.36) and (3.37).

All the previous results involving the typical cell remain valid for this modification of themodel. In particular, the results of Proposition 8 can be extended to the model with shadowing(where the cell associated to each base station is not necessarily the Voronoi cell) provided theorigin 0 belongs to a unique cell almost surely.

Note that the mean cell surface E0[|V (0)|] = 1/γ, and hence the mean traffic demand per cellE0[ρ(0)] = ρ/γ, do not depend at all on the shadowing. The values of other characteristics of thetypical and the mean cell will change depending on the distribution of the random shadowingfield Sv(y) (both the marginal distributions and the correlation across y). An interesting remarkin this regard is as follows.

In the full interference model, the mean load of the typical cell and the load of the meancell (which are by the definition equal E0[θ(0)] = θ) depend only on the stationary marginaldistribution of SINR(0,Φ), cf. (3.40). Hence, it does not depend on the (spatial) correlationof Sv(y) across y. Moreover, this distribution is known in the case of the Poisson networkand identically distributed marginal shadowing Sv(y) ∼ S. As explained in [19], in this caseSINR(0,Φ) in the model with shadowing has the same distribution as in the model withoutshadowing and the density of stations equal to γ ×E[S2/β ]. In particular, a specific distributionof S and the correlation of Sv(y) across y play no role. This equivalence of the two models (withand without shadowing) is more general, as explained in [19] and [21], and applies also to themean-cell-load equation (3.37).

Remark 19 The impact of the shadowing on the mean cell model, both in full and weightedinterference scenario in the above Poisson model with Sv(y) ∼ S can be summarized as fol-lows. It modifies only the cell load θ and not the traffic demand ρ. Moreover, multiplying γ byE[S2/β ] and dividing ρ by the same moment one obtains an equivalent (in terms of all consideredcharacteristics) mean cell without shadowing.

3.5.4 Mean cell

It is tempting to look for a synthetic model which would allow to relate main parameters and QoSmetrics of a large irregular cellular network in a simple, yet not simplistic way. The typical cellapproach described up to now offers such possibility. In this section we will go a little bit furtherand propose an even simpler model mimicking that of Section 3.4.1. It consists in considering avirtual cell, to which we will assign the parameters and QoS metrics inspired by the analysis ofthe typical cell. In contrast to the typical cell, our virtual cell is not random and this is why wecall it the mean cell. Specifically, we define it as a (virtual) cell having the same traffic demandρ and load θ as the typical cell. Note that these two characteristics admit explicit expressions;


cf. Proposition 8 and Remark 14.

ρ := E0 [ρ (0)] =ρ

γ, (3.39)

θ := E0 [θ (0)] =ρ

γE[1/R (SINR (0,Φ))] . (3.40)

For the remaining characteristics, we assume that they are related to the above two via therelations presented in Section 3.5.2. Specifically, following (3.20) we define the area of the meancell by V = ρ/ρ and in analogy to (3.25) we define the critical load of the mean cell as

ρc :=ρ

θ. (3.41)

We say that the mean cell is stable if ρ < ρc. Inspired by (3.22) we define the user’s throughputin the mean cell by

r := max (ρc − ρ, 0)

and, as in (3.23), the mean number of users in the mean cell is defined as

N :=ρ

r.

We observe the following immediate relations.

Corollary 4 The mean cell is stable if and only if θ < 1. In this case

N =θ

1− θ, (3.42)

r = ρ(1/θ − 1) , (3.43)

which are analogous to (3.27) and (3.28), respectively.

Remark 20 The equation (3.43) provides an alternative macroscopic relation between the trafficdemand and the mean user throughput in the network. It is purely analytic; no simulations arerequired provided one knows the distribution of the SINR of the typical user in (3.40). It willbe validated by comparison to real data measurements. We consider it as an approximationof (3.33). It consists in assuming N ' N0/πS . This latter hypothesis will be also separatelyvalidated numerically.

Remark 21 Note that the key characteristic of the mean cell is its load θ. In analogy to the loadfactor of the (classical) M/G/1 processor sharing queue, it characterizes the stability condition,mean number of users and the mean user throughput.

3.6 Heterogeneous networks

The objective of the present section is to extend our model to heterogenous wireless cellularnetworks comprising different categories of base stations transmitting at distinct powers. Thismodel permits to calculate by static simulation (Monte carlo estimation of some functionals ofPoisson point process) the quality of service perceived by the users served by each category ofbase stations. Analytical approximations are also proposed.

Specifically, we shall build a model for cellular networks comprising macro and micro BSemitting different powers. The user is served by the BS offering the strongest received power

3.6. HETEROGENEOUS NETWORKS 61

among all the BS in the network. In this context, we calculate the quality of service (QoS) offeredto the users by each category of BS (macro and micro). Extending the results from Section 3.5we develop here also the corresponding typical cell and mean cell approach globally and for eachbase station category. Further, we analyse the QoS parameters: mean user throughput, meannumber of users and cell load per base station category.

3.6.1 Model description

We consider a cellular network comprising different categories of BS characterized by differenttransmitting powers. In this section we restrict ourselves to a Poisson model. Let O be the finiteset of possible categories of base stations. The locations of BS of category o ∈ O are modeledby a homogeneous Poisson point process of intensity parameter γo ∈ R+ and denoted by Φo.The transmitting power of BS of category o is denoted by Po ∈ R+. Let Φ be the superpositionof Φoo∈O. Then Φ = vnn∈N is a homogeneous Poisson point process of intensity parameterγ =

∑o∈O γo.

Inversely, starting from the point process Φ we may retrieve the processes with the samedistribution as Φoo∈O in the following manner. Let Zn be i.i.d. marks of Φ such that

P(Zn = o) =γoγ, o ∈ O,n ∈ N∗

ThenΦo =

∑n∈N∗

δvn1 Zn = o , o ∈ O

Much as in Section 3.5 the propagation loss due to distance is a power function l(x) = (K |x|)βwhere K > 0 and β > 2 are given constants. The shadowing between a given station vn ∈ Φ andall locations y ∈ R2 is modeled by some stochastic process Sn (y − vn) taking values in R+. Theshadowing stochastic processes Sn (·) are i.i.d. marks of Φ. We assume that S1(y) are identicallydistributed across y. (We do not make any assumption concerning the dependence of S1(y) acrossy.) Thus the received power at location y ∈ R2 from base station vn ∈ Φ equals

PZnSn (y − vn)

l (|y − vn|)

The shadowing fields Sn (·) and the categories of BS Zn are assumed independent.In order to simplify the notation, let Z (x)x∈R2 be a stochastic process such that Z (vn) =

Zn. In a similar way, let Sx (·)x∈R2 be a stochastic process such that Svn (·) = Sn (·).Let Lvn (y) be the inverse of the received power at location y from BS vn; that is

Lvn (y) =l (|y − vn|)

PZ(vn)Svn (y − vn), n ∈ N∗ (3.44)

Each user is served by the BS offering the strongest received power among all the BS in thenetwork. Then the cell served by BS u ∈ Φ is

V (u) =y ∈ R2 : Lu (y) ≤ Lv (y) ,∀v ∈ Φ

(3.45)

The signal to interference and noise ratio (SINR) at location y in the downlink (BS to user)equals

SINR (y,Φ) =

1Lu(y)

N +∑v∈Φ\u

1Lv(y)

, y ∈ V (u) , u ∈ Φ (3.46)


where N is the noise power. Note that the above expression of SINR relies on the full interferencemodel from Definition 1 which will be improved in Section 3.6.4.

We assume that the peak bit-rate at location y, defined as the bit-rate of a user located at ywhen served alone by its BS, is some function R(SINR) given for example in (3.17). Local cellcharacteristics and traffic dynamics are assumed to be the same as in Section 3.5.2.

3.6.2 Typical and mean cell in multi-tier network

We aim to study the distribution of the received powers for a user located at the origin. Theinverse of the power received from BS vn ∈ Φ may be deduced from (3.44)

Ln := Lvn (0) =l(|vn|)SnPZn

where Sn := Sn (0− vn) are the shadowing random variables. Let Φ = Lnn∈N∗ which is apoint process on R+ and S = S1.

Lemma 5 Assume that E[S2/β

]< ∞. Then Φ = Lnn∈N∗ is a Poisson point process with

intensity measureΓ (0,m] = am2/β

where

a :=πE[S2/β

]K2

∑o∈O

γoP2/βo

Proof. It follows from our assumptions on the shadowing that the random variables Sn =Sn (−vn) are i.i.d. marks of Φ. Introducing Sn := SnPZn , we get

Ln =l(|vn|)Sn

Since Sn := SnPZn are i.i.d. marks of Φ (which may be viewed as modified shadowing), then Φmay be obtained by a transformation of Φ through the kernel

p(x,A) = P(l(x)

S1

∈ A)

By the displacement theorem Φ is a Poisson process with intensity measure

Γ [0,m) =

∫R2

p(x, [0,m))γdx

= γ

∫R2

P(l(x)

S1

∈ [0,m)

)dx

= γ

∫R2×R

1

l(x)

s∈ [0,m)

dxPS1

(ds)

= γ

∫R2×R

1

|x| < (sm)

1β

1

K

dxPS1

(ds)

= γ

∫R

π (sm)2/β

K2PS1

(ds) =γπ

K2E[S

2/β1

]m2/β


Note thatE[S

2/β1

]= E

[S2/β

]E[P

2/βZn

]= E

[S2/β

]∑o∈O

γoP2/βo

which completes the proof.It follows from the above lemma that the typical user (located at the origin) receives powers

from the different BS as if the network was a homogeneous one, i.e. with a single category ofBS transmitting at power

P =

(∑o∈O

γoγP 2/βo

)β/2(3.47)

Let Φo be the process of the inverses of the received powers from BS of category o ∈ O. Theyare independent Poisson point processes with respective intensity measures

Γo (0,m] = aom2/β , o ∈ O

where

ao =πE[S2/β

]K2

γoP2/βo , o ∈ O

We may view the whole process Φ as a superposition of the point processes

Φo

o∈O

. Inversely,

these latter processes may be obtained from Φ = Lnn∈N∗ by generating i.i.d. marksZn

n∈N∗

such thatP(Zn = o) =

aoa, o ∈ O,n ∈ N∗ (3.48)

in which caseΦo =

∑n∈N∗

δLn1Zn = o

, o ∈ O

Let Z∗ be the category of the BS serving the user located at the origin and L∗ = minn∈N∗ Ln.

Lemma 6 The probability that the user at the origin is served by a BS of category o ∈ O equals

P (Z∗ = o) =aoa

(3.49)

Moreover, Φ and Z∗ are independent and in particular the random variables L∗ and Z∗ areindependent.

Proof. We may assume without loss of generality that the points Lnn∈N∗ of Φ are sortedin the increasing order. Then

P (Z∗ = o) = P(Z1 = o

)=aoa

On the other hand, since Z∗ = Z1, it follows from (3.48) that Z∗ is independent from Φ.The above lemma shows that the process of powers received from all the BS is independent

from the category of the serving base station. In particular, the interference and the signal tointerference and noise ratio (SINR) are independent from the category of the serving base station.

Remark 22 It follows from Lemma 6 that P (L∗ > m|Z∗ = o) = P (L∗ > m); i.e. the distribu-tion of the received power from each category of BS is the same.


3.6.3 Full interference model

The average of the characteristics of the cells described in Section 3.6.1, i.e. Section 3.5.2 over thewhole network gives a first global indication of the network performance. We may also averagethese characteristics per category of BS leading to the notion of typical cell for each categoryof BS. The objective of the present section is to establish the relations between these averages.The Section title relates to the load (interference) scenarios in 1.

Typical cell approach

Traffic and load We shall average the characteristics of the cells over an increasing sequence ofdiscs, denoted by A, of radii going to infinity, exactly in the same way as it is done in Section 3.5.3;for example the global average traffic demand per cell equals

ρ := lim|A|→∞

1

Φ (A)

∑v∈Φ∩A

ρ (v)

whereas the average traffic demand per cell of category o is

ρo := lim|A|→∞

1

Φo (A)

∑v∈Φo∩A

ρ (v) (3.50)

We define similarly the global average load θ and the average load θo for cells of category o.The following lemma gives the explicit expressions of the average traffic demands per cell

both globally and for each category of BS. Palm theory will be useful in the proof of this lemmaas well as for upcoming results. The following statements are also inspired by the considerationsin Section 3.5.3. In particular we shall make use of the Palm probability P0 associated to thepoint process Φ of base station locations. Let E0 [·] be the expectation with respect to P0.

Lemma 7 We have

ρ =ρ

γ

ρo =ρaoγoa

, o ∈ O (3.51)

Proof. Since the point process Φ is ergodic, it follows from the ergodic theorem for pointprocesses that [36, Proposition 12.2.VI]

ρ = E0 [ρ (0)] = ρE0 [|V (0)|] =ρ

γ

where the second equality is due to (3.20), and the last equality follows for the inverse formula ofPalm calculus [9, Theorem 4.2.1] (which may be extended to the case where the cell associatedto each BS is not necessarily the Voronoi cell; the only requirement is that the user located at 0belongs to a unique cell almost surely). Similarly,

ρo = lim|A|→∞

Φ (A)

Φo (A)

1

Φ (A)

∑v∈Φ∩A

ρ (v) 1 v ∈ Φo

=γ

γoE0 [ρ (0) 1 0 ∈ Φo]

= E0 [ρ (0) |0 ∈ Φo] = ρE0 [|V (0)| |0 ∈ Φo] (3.52)


Moreover,

E0 [|V (0)| |0 ∈ Φo] =E0 [|V (0)| × 1 0 ∈ Φo]

P0 (0 ∈ Φo)

=

1γP (v∗ ∈ Φo)

P0 (0 ∈ Φo)

=

1γaoa

γoγ

=aoγoa

(3.53)

where the second equality follows from the inverse formula of Palm calculus, v∗ designates theBS serving the origin and the third equality follows from Lemma 6.

Remark 23 We may interpret Equation (3.53) as follows. Consider a sufficiently large area Aand denote its area by |A|. By the ergodic theorem, there are γo |A| base stations of category owithin A. On the other hand, the probability that a user is served by a BS of category o is ao

aand consequently a portion ao

a of the area of A is covered by γo |A| base stations. So, the meancell area of a base station of category o is

aoa |A|γo |A|

=aoγoa

The following lemma shows that the global average cell load is related to the stationarydistribution of the SINR for a user located at the origin. It gives also the expression of theaverage load for cells of each category.

Proposition 9 The average loads per cell equal

θ =ρ

γE[R−1 (SINR (0,Φ))

](3.54)

θo = θγaoγoa

= θP

2/βo

P 2/β, o ∈ O (3.55)

where P is given by (3.47).

Proof. Along the same lines as the proof of (3.52), we have

θo = E0 [θ (0) |0 ∈ Φo]

On the other hand, let

g (y,Φ (ω)) = ρR−1 (SINR (y,Φ (ω))) , y ∈ R2, ω ∈ Ω

let v∗y be the BS serving the user at location y, and let v∗ be the BS serving the origin, then

θo = E0 [θ (0) |0 ∈ Φo]

= E0 [θ (0) 1 0 ∈ Φo] /P0 (0 ∈ Φo)

= E0

[∫V (0)

g (y,Φ) 1v∗y ∈ Φo

dy

]/P0 (0 ∈ Φo)

=1

γE [g (0,Φ) 1 v∗ ∈ Φo] /P0 (0 ∈ Φo)

=1

γE [g (0,Φ)]

P (v∗ ∈ Φo)

P0 (0 ∈ Φo)=

1

γE [g (0,Φ)]

γaoγoa


where the fourth equality follows from the inverse formula of Palm calculus [9, Theorem 4.2.1],the fifth equality follows from Lemma 6 and the last equality follows from (3.49). It follows that

θ = E0 [θ (0)]

=∑o∈O

E0 [θ (0) |0 ∈ Φo]P0 (0 ∈ Φo)

=∑o∈O

1

γE [g (0,Φ)]P (v∗ ∈ Φo) =

1

γE [g (0,Φ)]

Remark 24 Heterogeneous versus homogeneous network. Surprisingly, the global average cellload in the heterogeneous network is the same as that in the corresponding homogeneous network(where all the BS transmit the same power (3.47)). This observation holds for the current fullinterference model.

Users number and throughput Note that if for BS v, θ (v) ≥ 1 then the correspondingnumber of users N (v) is infinite; in which case we say that BS v is instable. The empiricalaverage of the number of users over the cells in the network would then be infinite. In order toavoid this degeneration, we define the global average number of users as

N := lim|A|→∞

1

Φ (A)

∑v∈Φ∩A

N (v) 1 θ (v) < 1

= E0 [N (0) 1 θ (0) < 1]

where the second equality follows from ergodicity. Similarly, the average number of users for BSof category o ∈ O

No := lim|A|→∞

1

Φo (A)

∑v∈Φo∩A

N (v) 1 θ (v) < 1 (3.56)

= E0 [N (0) 1 θ (0) < 1 |0 ∈ Φo]

Let So be the union of stable cells of category o ∈ O; that is

So =⋃

v∈Φo:θ(v)<1

V (v)

and S =⋃o∈O So. The user’s average throughput is defined as the mean transmitted volume

per call; i.e. 1/µ, divided by the mean call duration; that is

r := lim|A|→∞

1/µ

mean call duration in A ∩ S

and for category o ∈ O,

ro := lim|A|→∞

1/µ

mean call duration in A ∩ So


Proposition 10 We have

r =ρ

Nπ

ro =ρoNo

πo, o ∈ O (3.57)

where

π = P (θ (v∗) < 1)

πo = P (θ (v∗) < 1|v∗ ∈ Φo) , o ∈ O

where v∗ is the BS serving the origin.

Proof. Let Wo =⋃v∈A∩So V (v). Consider call arrivals and departures to Wo over a suffi-

ciently large time interval. Let TWo be the mean call duration in Wo, and let NWo be the meannumber of users in Wo. By Little’s law

NWo = λ |Wo|TWo

Thus the user’s throughput in Wo equals

1/µ

TWo=ρ |Wo|NWo

= ρ

∑v∈A∩So |V (v)|∑v∈A∩So N (v)

= ρ

∑v∈A∩Φ |V (v)| 1 θ (v) < 1, v ∈ Φo∑v∈A∩ΦN (v) 1 θ (v) < 1, v ∈ Φo

When |A| → ∞, it follows from the ergodic theorem that

ro = ρE0 [|V (0)| 1 θ (0) < 1, 0 ∈ Φo]E0 [N (0) 1 θ (0) < 1, 0 ∈ Φo]

On the other hand, it follows from the inverse formula of Palm calculus that

E0 [|V (0)| 1 θ (0) < 1, 0 ∈ Φo] =1

γP (θ (v∗) < 1, v∗ ∈ Φo)

Then

ro =ρ

γ

P (θ (v∗) < 1, v∗ ∈ Φo)

P0 (0 ∈ Φo) No

=ρ

γ

P (v∗ ∈ Φo)P (θ (v∗) < 1|v∗ ∈ Φo)

P0 (0 ∈ Φo) No

=ρoNo

P (θ (v∗) < 1|v∗ ∈ Φo) =ρoNo

πo

The expression for r may be proved in the same lines as above.Under Palm probability, the cell of the base station located at 0 is usually called typical cell.

Note that the typical cell has not a concrete existence, but it is rather a useful mathematical toolto analyze or predict the behavior of the empirical averages over many cells. We define both a


global typical cell and also a typical cell per category of BS. The global typical cell has a trafficdemand ρ, a load θ, a users number N and a user’s throughput r. The typical cell of categoryo ∈ O has a traffic demand ρo, a load θo, a users number No and a user’s throughput ro.

Unfortunately, neither the number of users nor the user’s throughput in the typical cell haveexplicit analytic expression. We shall introduce in the following section an approximation calledmean cell whose characteristics admit explicit expressions and approximate those of the typicalcell both globally and for each category of BS.

Mean cell approximation

We define a mean cell as a virtual cell having traffic demand ρ := ρ and load θ := θ. Similarly,we introduce a mean cell for each category o ∈ O as a virtual cell having traffic demand ρo := ρoand load θo := θo. The mean cells are assumed to behave like a concrete cell from the queuingtheory point of view as described in Sections 3.6.1 and 3.5.2. Specifically, we define the criticalload of each mean cell in analogy to (3.25) as

ρc :=ρ

θ, ρco :=

ρo

θo, o ∈ O

Inspired by (3.22), the user’s throughput in the mean cell is defined as

r := max(ρc − ρ, 0), ro := max(ρco − ρo, 0), o ∈ O

The users number in the mean cell is defined following (3.23) as

N =ρ

r, No =

ρoro, o ∈ O

Combining Equations (3.51) and (3.55), we see that ρco = ρc; that is the critical traffic of themean cell is the same for all the categories of the BS. Moreover, observe that all the characteristicsof the mean cell are straightforwardly deduced from its traffic and load which in view of (3.54)depends on the distribution of SINR (0,Φ) whose analytic expression is given in [19].

We shall evaluate the mean cell approximation (both globally and per category) by compar-ison to the characteristics of the typical cell obtained both from simulation and from real fieldmeasurements.

3.6.4 Weighted interference model

Typical cell approach

The expression (3.46) of SINR relies on the assumption that the interfering BS are always trans-mitting at their maximal power. In real networks, the BS transmits only when it serves at leastone user. Since the probability of such event is given by (3.24), we modify the expression ofSINR as follows, for y ∈ V (u) , u ∈ Φ,

SINR (y,Φ) =

1Lu(y)

N +∑v∈Φ\u

min(θ(v),1)Lv(y)

(3.58)

It follows from Equation (3.25) that

θ (u) = ρ

∫V (u)

R−1

1Lu(y)

N +∑v∈Φ\u

min(θ(v),1)Lv(y)

dy (3.59)

3.7. NUMERICAL RESULTS: NETWORK DIMENSIONING AND QOS ESTIMATION 69

which is a system of equations with the loads θ (v)v∈Φ as unknowns. Given a realizationof the network, one gets the loads of the different BS by solving numerically the above system,similarly as in 3.5.3. The traffic demand (or equivalently surface) of each cell ρ (v)v∈Φ may alsobe estimated numerically. The other characteristics of each cell are then deduced from the loadand traffic demands using the relations in Section 3.6.1 and 3.5.2; specifically, the critical trafficis deduced from (3.25), the user’s throughput from (3.22) and the users number from (3.23).

The characteristics of the typical cell for each category of BS may then be computed by takingthe empirical averages over the cells of each category as explained in Section 3.6.3. Specifically,the traffic is calculated by (3.50) with a similar formula for the load, the number of users iscalculated by (3.56) and the user’s throughput is given by (3.57) since Proposition 10 holdstrue in the present context. Moreover, the results of Proposition 9 remain true; in particularthe load of the typical cell of a given category θo is related to the global typical cell load θ byEquation (3.55).

We will see in the numerical section that such model, called (load-)weighted interferencemodel, fits better to real field measurements than the full interference model. In particular, thefact that the average load of each category of BS increases with its power through a simple lawwill be validated by real field measurements in the numerical section.

Mean cell approximation

In the mean cell approximation, the expression (3.58) of the SINR is modified by replacing theload of each BS by the mean load of the corresponding category; that is

SINR (y,Φ) =

1Lu(y)

N +∑o∈O θo

∑v∈Φo\u

1Lv(y)

Moreover, by analogy to (3.54) and (3.55) we assume that

θ =ρ

γE[R−1

(SINR (0,Φ)

)]θo = θ

P2/βo

P 2/β(3.60)

Combining the three above equations we deduce that θ is solution of the following fixed-pointequation

θ =ρ

γE

[R−1

(1

Lu(0)

N + θ∑o∈O αo

∑v∈Φo\u

1Lv(0)

)](3.61)

where

αo :=P

2/βo

P 2/β, o ∈ O (3.62)

Solving the above equation we get the load of the mean cell; then we deduce the load of themean cell of each category by applying (3.60). The characteristics of the mean cell both globallyand for each category are then deduced from its traffic and load by the relations presented inSection 3.6.3.

3.7 Numerical results: network dimensioning and QoS es-timation

In this Section the following results are presented:


• Firstly, the validation is done for a dynamic context of users arivals and departures fromthe network by comparison to 3GPP simulations results.

• Then, the results for dynamic context are applied to dimension hexagonal cellular LTEnetwork based on Section 3.4.

• Finally, the previous two tasks are extended to demonstrate the results concerning irreg-ular networks. QoS estimation as function of traffic demand is done for irregular cellularnetworks and results are compared to real-field measurements.

Throughout this Section we assume that user channels are intra-cell orthogonal and inter-cellindependent : if BS u serves n users located at y1, y2, . . . , yn ∈ V (u) then the bit-rate of theuser located at yj equals to 1/n th of its peak bit-rate 1

nR (SINR (yj ,Φ)), j ∈ 1, 2, . . . , n. 7Thepattern of BS Φ does not evolve in time.

3.7.1 Validation in a dynamic context

The aim is to compare the results of the queueing approach described in Section 3.4.1 to those of3GPP simulations in a dynamic context ; i.e. calls arrive and depart from the network and eachbase station transmits only when it has at least one user to serve. This context is called FTPtraffic model in [3, §A.2.1.3.1].

As simulation results, we consider the results of tools which are compliant with 3GPP ap-proach [3] (an Orange simulator developed in C++ being one of them). The average (as wellconfidence intervals at 20% and 80%) of the results of the different contributors to 3GPP will beplotted and compared to our analytical approach (implemented in Matlab).

We begin by describing the subset of the parameters in [1, Table A.2.1.1-3], [3, Table A.2.2-1]which are used in our analytical calculations. The frequency carrier equals 2GHz; the path-lossmodel is l (r) = 128.1 + 37.6 × log10(r) [in dB] (where r is in km); the penetration loss equals20dB and the shadowing is centered and log-normally distributed with standard deviation 8dB.

The antenna pattern in the horizontal plane is A (ϕ) = −min(

12 (ϕ/ϕ3dB)2, Am

)where ϕ3dB =

70, Am = 20dB. The system bandwidth equals W = 10MHz; the noise power is N = −95dBmand the base station transmission power equals P = 60dBm (including antenna gain).

Figure 3.1 gives the load as function of traffic demand per cell resulting from 3GPP simulationsand from the queueing approach. For 3GPP simulations, the load is calculated as the fractionof time where a base station has at least one user to serve. The average of the simulation resultsof the different 3GPP contributors as well the confidence intervals at 20% and 80% are plotted.For the queueing approach, the load is calculated by Equation (3.19) where the critical trafficρc is the solution of the fixed-point problem (3.18). We observe in Figure 3.1 that the two loadscalculated by these two methods are close, except when the queueing load is close to 1. Indeedin this case, the system is at its limit of stability and therefore the time averages converge veryslowly to their ergodic limits [80, p.114]. This explains why the 3GPP simulations are too timeconsuming at high loads (up to 3 weeks of calculation) and also the gap between the simulationand queueing loads in Figure 3.1. To get the curves in Figure 3.1, the computing time for the3GPP simulations is several weeks whereas it is about 1 minute for the queueing approach.

Figure 3.2 gives the mean user throughput as function of traffic demand for the different loadsituations described in Definition 1 and for 3GPP simulations. In this latter case, the mean userthroughput is obtained by averaging the users throughput over the whole simulation time.

7This can be achieved using various multiple access schemes, e.g. time (related to HSDPA) or frequency(related to LTE) division.

3.7. NUMERICAL RESULTS 71

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000 10000

Load

Traffic demand per cell [kbit/s]

3GPP simulations

Queueing approach

Difference

Figure 3.1: Load versus traffic demand per cell

One can observe that the curve obtained from 3GPP simulations and the one derived fromthe queueing approach with adapted load are close, except for the highest value of traffic demand(which corresponds to a load close to 1). This gap is due to the slow convergence rate of the3GPP simulations discussed above. On the other hand, as expected, the curves for the full andadapted load converge for the highest value of the traffic demand since this corresponds to thelimit of stability of the network (when the user throughput vanishes). Going backward withthe values of traffic demand, the difference between these two curves increases up to factor 5.Finally, the null and adapted load curves have the same value for the smallest traffic demandand diverge as the traffic demand increases.

Figure 3.3 gives the 95% quantile of user throughput as function of the traffic demand for3GPP simulations and the queueing approach. Observe that the quantiles of the 3GPP simula-tions are smaller than those of the queueing approach with adapted load; nevertheless the twocurves have the same tendency. This is related to the fact that peak bit-rates of 3GPP simula-tions are more dispersed than the analytical ones as shown in Figure 2.5. On the other hand,the null and full load curves agree with the adapted load one for the smallest and the highestvalues of traffic demand, respectively.

In this Section we want to use all previously developed results and apply them to variousnumerical examples of cellular network dimensioning and QoS performance evaluation. We willuse the results from Section 3.4 to calculate the cell radius necessary to satisfy the traffic demandand some QoS constraint with a given network setup, which is network dimensioning.

Using the model described in Section 3.5 we will estimate the QoS parameters and networkperformance and compare the results to the real-field measurements.


0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 2000 4000 6000 8000 10000

(Arit

hmet

ic m

ean)

Use

r th

roug

hput

[kbi

t/s]


3GPP simulations

Queueing approach, Null load

Queueing approach, Adapted load

Queueing approach, Full load

Figure 3.2: Mean user throughput versus traffic demand per cell

0

10000

20000

30000

40000

50000

60000

0 2000 4000 6000 8000 10000

(95%

Qua

ntile

) U

ser

thro

ughp

ut [k

bit/s

]


3GPP simulations

Queueing approach, Null load

Queueing approach, Adapted load

Queueing approach, Full load

Figure 3.3: 95% quantile of user throughput versus traffic demand

3.7.2 Hexagonal LTE network dimensioning

We aim now to illustrate the processor-sharing queueing approach for cellular network dimen-sioning purposes applied to a multi-cell scenario in symmetric hexagonal networks. In this caseall cells are ”the same”, that is to say, if we take randomly any cell then it represents the typicalcell, so we can use the approach described in Section 3.4 for network dimensioning. Figure 3.4shows mean user throughput as function of cell radius for traffic demand densities equal to 0.1and 10Mbit/s/km2 and different load situations (see Definition 1). The mean user throughput ris calculated by (3.8) where the critical traffic ρc is the solution of the fixed-point problem (3.18).

As expected, for each value of the traffic demand, the curves are in decreasing order forrespectively the null, adapted and full load situations. Moreover, observe that the null andadapted load curves are close to each other for the small value of traffic demand since in thiscase the interference is too small. Contrarily, for higher value of traffic demand, interference is


0

20000

40000

60000

80000

100000

120000

0 0.5 1 1.5 2 2.5 3 3.5

(Arit

hmet

ic m

ean)

Use

r th

roug

hput

[kbi

t/s]

Cell radius [km]

Traffic=0.1 Mbit/s/km2, Null loadTraffic=0.1 Mbit/s/km2, Adapted load

Traffic=0.1 Mbit/s/km2, Full loadTraffic=10 Mbit/s/km2, Null load

Traffic=10 Mbit/s/km2, Adapted loadTraffic=10 Mbit/s/km2, Full load

Figure 3.4: Mean user throughput as function of the cell radius for different load situations

significant so that adapted load curve deviates from the null load one. Moreover, as observedpreviously, the adapted and full load curves converge for the limit of stability of the network(when user throughput vanishes). The computing time to get Figure 3.4 is some minutes, whereasit would require several weeks for 3GPP simulations.

Figure 3.5 shows mean user throughput as function of cell radius for different traffic demanddensities for adapted load situation. As expected each curve is decreasing and ultimately van-ishes for some critical value of cell radius corresponding to the stability limit of the network.Additionally, when the traffic increases, the curves decrease and the critical cell radius decreasesrapidly. On the other hand, the curves for traffic demands of 10 and 100kbit/s/km2 are close tothat of null traffic up to the cell radius of 2km which shows that noise is preponderant againstinterference.

0

20000

40000

60000

80000

100000

120000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(Arit

hmet

ic m

ean)

Use

r th

roug

hput

[kbi

t/s]

Cell radius [km]

Traffic=0 kbit/s/km2

Traffic=10 kbit/s/km2

100 kbit/s/km2

1 Mbit/s/km2

10 Mbit/s/km2

Figure 3.5: Mean user throughput as function of the cell radius for different traffic demanddensities (adapted load)


We give now the numerical solution of the dimensioning problem in terms of the cell radiuswhich is more appealing than the number of base stations per unit surface (note, however, thatthe latter is inversely proportional to the square of the former). Figure 3.6 shows cell radiusversus traffic demand density for two target arithmetic means of the user throughput equal to1 and 10Mbit/s for the three load situations described in Definition 1. For the smaller userthroughput, the three curves are close to each other whereas for the larger throughput theydiffer significantly from each other. The adapted load curve lies between the null and full loadones; and meets each of them for low and high traffic, respectively. This is due to the fact thatwhen traffic increases, the network evolves from a noise-limited to an interference-limited regime.

0

1

2

3

4

5

6

1 10 100 1000 10000 100000

Cel

l rad

ius

[km

]

Traffic demand density [kbit/s/km2]

User throughput=1 Mbit/s, Null loadUser throughput=1 Mbit/s, Adapted load

User throughput=1 Mbit/s, Full loadUser throughput=10 Mbit/s, Null load

User throughput=10 Mbit/s, Adapted loadUser throughput=10 Mbit/s, Full load

Figure 3.6: Cell radius versus traffic demand density for mean user throughput 104kbit/s

Figure 3.7 shows cell radius versus traffic demand density for different target values of thearithmetic mean of the user throughput. As expected, the cell radius is decreasing with thetraffic demand and with the user throughput. Note that, for the three largest user throughputs,the curve comprises a stationary part corresponding to a coverage constraint and a decreasingpart corresponding to a capacity constraint.

Figures 3.6 and 3.7 are obtained in few minutes by the analytical approach, whereas theywould require several months for 3GPP simulations.


0

1

2

3

4

5

6

7

1 10 100 1000 10000 100000

Cel

l rad

ius

[km

]

Traffic demand density [kbit/s/km2]

User throughput=100 kbit/s1 Mbit/s5 Mbit/s

10 Mbit/s50 Mbit/s

Figure 3.7: Cell radius versus traffic demand density for different mean user throughputs

3.7.3 Mean performance estimation of irregular network using Poissonprocess

Here, we consider the irregular cellular network with orthogonal users’ channels as described inSection 3.5 and produce the examples showing the network performance as function of the trafficdemand. Also, the figures showing the mean user throughput as function of the traffic demandcan be useful not only in the sense of their own explicit importance (gives the information aboutthe QoS in a network), but also for network dimensioning and capacity.

To illustrate the motivation of this work, we present first in Figure 3.8 non-averaged dataobtained from the measurements performed in an operational network in some zone of somebig city in Europe. More precisely, a dense urban network zone consisting of 382 base stationswas selected in a big European city, whose locations loosely satisfy the homogeneous spatialPoisson assumption. Ripley’s L-function, cf [88, page 50], plotted on Figure 3.16, was used toverify this latter assumption. The density of base stations in this dense urban zone is about 4.62base stations per km2. Later, we will consider also an urban zone of a different European city,where the spatial homogeneous Poissonianity of the base station locations can also be retained;cf. Figure 3.16, with roughly four times smaller density of base stations, more precisely 1.15stations per km2. In both cases the network operates a HSDPA system with MMSE coding.

Different points in this figure correspond to the measurements of the traffic demand and theestimation of the user throughput made by different cells during different hours of the day. Noapparent relation between these two quantities can be observed in this way.

In order to uderstand and predict the performance of the network for which we have presentedthe above data, we will now specify correspondingly our general model and study it using theproposed approach. The obtained results will be compared to the appropriately averaged realfield measurements.

Consider the following numerical setup. Assume a Poisson process of BS with intensityγ = 4.62km−2 (which corresponds to an average distance between two neighbouring BS of0.5km). We assume the path-loss function l(r) = (Kr)β , with K = 7117km−1, and the pathloss exponent β = 3.8. The propagation model comprises the log-normal shadowing with thelogarithmic standard deviation 10dB; cf [20], and the mean spatial correlation distance 0.05km.

The transmission power is P = 58dBm, with a fraction ε = 10% used in the pilot channel.


Figure 3.8: Local user throughput versus local traffic demand for some zone (selected to satisfya spatial homogeneity of the base stations) of an operational cellular network deployed in a bigcity in Europe. 9288 different points correspond to the measurements made by different sectorsof different base stations during 24 different hours of some given day.

The antenna pattern is described in [3, Table A.2.1.1-2]. The noise power is N = −96dBm.

We assume the peak bit-rate to be equal to 30% of the ergodic capacity of the AWGN channelwith the frequency bandwidth W = 5MHz and the Rayleigh fading with mean power E[|H|2] = 1.

Estimations of the typical cell are performed by the simulation of 30 realizations of the Poissonmodel within a finite observation window, which is taken to be the disc of radius 2.63km. Wefirst average over all BS in this window and then over the model realizations. The empiricalstandard deviation from the obtained averages will be presented via error-bars.

We shall study now our model using the typical and mean cell approach, assuming first thefull interference model and then the weighted one.

Full interference

We consider first the full interference model (i.e. all BS emit the signal all the time, regardless ofwhether or not they serve users). Figure 3.9 shows the mean cell load of the typical cell E0[θ(0)]and the stable fraction of the network πS obtained from simulations, as well as the analyticallycalculated load of the mean cell θ, versus mean traffic demand per cell ρ/γ. We confirm that thetypical cell and the mean cell models have the same load. Note that for a traffic demand up to500kbps per cell we do not observe unstable cells in our simulation window (πS = 1).

Figure 3.10 shows the mean number of users per cell in the stable part of the network N0/πS(obtained from simulations) and the analytically calculated number of users in the mean cellN versus mean traffic demand per cell. We have two remarks. For the traffic demand smallerthan 500kbps per cell (for which all the simulated cells are stable; πS = 1, cf. Figure 3.9), bothmodels predict the same mean number of users per cell. Beyond this value of the traffic demandper cell the estimators of the number of users in the typical cell become inaccurate due to thevery rapidly increasing fraction of the unstable region. (Error bars on all figures represent thestandard deviation in the averaging over 30 realizations of the Poisson network).

Finally, Figure 3.11 presents the dependence of the mean user throughput in the network


Figure 3.9: Cell load and the stable fraction of the network versus traffic demand per cell in thefull interference model.

Figure 3.10: Number of users per cell versus traffic demand per cell in the full interference model.

on the mean traffic demand per cell obtained using the two approaches: r0 and for the typicalcell and r for the mean cell. Again, both models predict the same performance up to roughly500kbps.


Figure 3.11: Mean user throughput in the network versus traffic demand per cell in the fullinterference model.

Weighted interference

We consider now the load-weighted interference model taking into account idling cells. We see inFigures 3.12, 3.13 and 3.14 that the consequence of this (more realistic) assumption is that thecell loads are smaller, a larger fraction πS of the network remains stable, and the two approaches(by the typical cell and by the mean cell) predict similar values of the QoS metrics up to a largervalue of the traffic demand per cell, roughly 700kbps. Note that it is in this region that the realnetwork operates for which we present the measurements, and that its performance coincides withthe performance metrics calculated using the typical and mean cell approach. More precisely,the field measurements in Figures 3.12, 3.13 and 3.14 correspond to the same day and networkzone considered in Figure 3.8.

Figure 3.12: Load and the stable fraction of the network versus traffic demand in the weightedinterference model. Also, load estimated from real field measurements.


Figure 3.13: Number of users versus traffic demand per cell in the weighted interference model.Also, the same characteristic estimated from the real field measurements.

Figure 3.14: Mean user throughput in the network versus traffic demand per cell in the weightedinterference model. Also, the same characteristic estimated from the real field measurements.

Remark 25 (Measurement methodology) Measurement points in Figure 3.12 show the frac-tion of time, within a given hour, when the considered base stations were idle, averaged over thebase stations, as function of the average traffic demand during this hour. Similarly, measurementpoints in Figure 3.13 show the spatial average of the mean number of users reported by the con-sidered base stations within a given hour, as function of the average traffic demand during thishour. Finally, measurements in Figure 3.14 give the ratio of the total number of bits transmittedby all the base stations during a given hour, to the total number of users they served during thishour as function of the average traffic demand during this hour.


Remark that Figure 3.14 makes evident a macroscopic relation between the traffic demandand the mean user throughput in the network zone already considered on Figure 3.8. Thisrelation, we are primarily looking for here, is not visible without the spatial averaging of thenetwork measurements described in Remark 25. In order to assure the reader that a relativelygood matching between the measurements and the analytic prediction is not a coincidence, wepresent in Figure 3.15 similar results for an urban zone of a different European city, wherethe spatial homogeneous Poissonianity of the base station locations can also be retained; cf.Figure 3.16. The only engineering difference of this network zone with respect to the previouslyconsidered dense urban zone is roughly four times smaller density of base stations, more precisely1.15 stations per km2.

Remark 26 (Day and night hours) Let us make a final remark regarding the empirical rela-tion between the mean user throughput and the mean traffic demand revealed in Figures 3.14 and3.15. Recall that different points in these plots correspond to different hours of some given day.In fact, the points below the mean curve correspond to day hours while the points above the meancurve correspond to night hours. This “circulation” of the measured values around the theoreticalmean curve, indicated in Figure 3.15 and visible in both presented plots of the throughput, seemsto be a more general rule, which escapes from the analysis presented in this thesis and remainsan open question. A possible explanation can lie in a different space-time structure of the trafficduring the day and night, with the former one being much more clustered (fewer users, requestinglarger volumes, generating less interference and overhead traffic).

Figure 3.15: Mean user throughput in the network versus traffic demand per area for an urbanzone of a big city in Europe. (The density of base stations is 4 times smaller than in the denseurban zone considered in Figure 3.14).


Figure 3.16: Ripley’s L-function calculated for the considered dense urban and urban networkzones. (L function is the square root of the sample-based estimator of the expected number ofneigbours of the typical point within a given distance, normalized by the mean number of pointsin the disk of the same radius. Slinvyak’s theorem allows to calculate the theoretical value of thisfunction for a homogeneous Poisson process, which is L(r) = r.) In fact, in large cities spatial,homogeneous “Poissonianity” of base-station locations is often satisfied “per zone” (city center,residential zone, suburbs, etc.). Moreover, log-normal shadowing further justifies the Poissonassumption, cf. [20, 29] .

3.7.4 Numerical results for heterogeneous networks

We consider the following numerical setting representative of an operational network in some bigcity in Europe comprising macro and micro base stations. The performance of each category ofBS calculated using the approach proposed in the present Chapter is compared to the real fieldmeasurements.

Model specification

The network comprises macro and micro base stations indexed by 1 and 2 respectively. The BSlocations are generated as a realization of a Poisson point process of intensity γ = γ1 + γ2 =4.62km−2 (which corresponds to an average distance between two base stations of 0.5km) over asufficiently large observation window which is taken to be the disc of radius 2.63km. The ratioof the micro to macro BS intensities equals γ2/γ1 = 0.039.

The powers transmitted by macro and micro BS equal P1 = 58.26dBm, P2 = 47.42dBmrespectively. The global average power calculated by (3.47) equals P = 58.03dBm.

The propagation loss due to distance is l(x) = (K |x|)β where K = 7117km−1 and the pathloss exponent β = 3.8. Shadowing is assumed log-normally distributed with standard deviationσ = 10dB and spatial correlation 0.05km.


The technology is HSDPA (High-Speed Downlink Packet Access) with MMSE (MinimumMean Square Error) receiver in the downlink. The peak bit-rate equals to 30% of the informationtheoretic capacity of the Rayleigh fading channel with AWGN; that is

R (SINR) = 0.3WE[log2

(1 + |H|2 SINR

)]where the expectation E [·] is with respect to the Rayleigh fading H of mean power E[|H|2] = 1,and W = 5MHz is the frequency bandwidth.

A fraction ε = 10% of the transmitted power is used by the pilot channel (which is alwaystransmitted whether the BS serve users or not). The antenna pattern is described in [3, TableA.2.1.1-2]. The noise power is −96dBm.

We study now our model using the typical and mean cell approaches, assuming first thefull interference model and then the weighted one. We shall give the performance results bothglobally for all the categories of BS and separately for macro and micro categories.

The results for the typical cell approach are obtained either by simulation or from measure-ments, whereas the results for the mean cell approach are analytic. Note that the mean load isknown analytically (3.19) for both the typical and mean cells; nevertheless, we will associate thisanalytic expression to the mean cell approach in the legends of the subsequent curves.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200

Cel

l loa

d

Traffic demand per cell [kbps]

Mean cell (homogeneous)Mean cell

Figure 3.17: Accuracy of the homogeneous approximation of the mean cell

Full interference

We consider first the full interference model (i.e. each BS always transmits at its maximal powereven when it has no user to serve). This model is analyzed by simulation with the typical cellapproach or analytically with the mean cell approach (no measurements are available in the fullinterference case).

Figure 3.18 shows the mean cell load of the typical cell θ and the stable fraction of the networkπ obtained from simulations, as well as the load of the mean cell θ calculated analytically, versusmean traffic demand per cell ρ/γ. This figure confirms that the typical cell and the mean cellmodels have the same load both globally and for each category of BS.

Figure 3.19 shows the mean number of users per cell N (obtained from simulations) and theanalytically calculated number of users in the mean cell N versus mean traffic demand per cell.


Again the mean cell reproduces well the results of the typical cell at least for moderate trafficdemands; i.e. as long as the stable fraction of the network π remains close to 1 as may be seenin Figure 3.18.

Finally, Figure 3.20 presents the dependence of the mean user throughput in the network onthe mean traffic demand per cell obtained using the two approaches: r and for the typical celland r for the mean cell. Observe again for moderate traffic demands the good fit between themean and typical cells both globally and for each BS category.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200

Cel

l loa

d

S

tabl

e fr

actio

n


Mean cell loadMacroMicro

Typical cell loadMacroMicro

Stable fractionMacroMicro

Figure 3.18: Cell load versus traffic demand per cell in the full interference model.

0

2

4

6

8

10

0 200 400 600 800 1000 1200

Num

ber

of u

sers


Mean cellMacroMicro

Typical cellMacroMicro

Figure 3.19: Number of users per cell versus traffic demand per cell in the full interference model.


0

500

1000

1500

2000

2500

3000

0 200 400 600 800 1000 1200

Use

r th

roug

hput

[kbp

s]


Mean cellMacroMicro


Figure 3.20: Mean user throughput in the network versus traffic demand per cell in the fullinterference model.

Weighted interference

We consider now the load-weighted interference model accounting for idle periods of the inter-fering BS.

Figure 3.21 shows the mean cell load and the stable fraction of the network, Figure 3.22presents the mean number of users per cell, and Figure 3.23 shows the mean user throughput.Besides the results of the analytic mean cell and the simulated typical cell, these figures givealso the typical cell characteristics deduced from measurements in the operational network. Asexpected, the performance is improved compared to the full interference model.

Moreover, observe that the analytic mean cell load and number of users fits well with boththe simulated and measured typical cell; particularly in the range of traffic demands for whichmeasurements are available. This agreement holds both globally and for each category of BS;in particular the dependence of the average load of each category of BS with its power in theoperational network is well explained by the simple law (3.60).


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000 1200

Cel

l loa

d

S

tabl

e fr

actio

n


Mean cell loadMacroMicro

Typical cell loadMacroMicro

Stable fractionMacroMicro

Field measurementsMacroMicro

Figure 3.21: Cell load versus traffic demand per cell in the weighted interference model.

0

2

4

6

8

10

0 200 400 600 800 1000 1200

Num

ber

of u

sers


Mean cellMacroMicro



Figure 3.22: Number of users per cell versus traffic demand per cell in the weighted interferencemodel.


0

1000

2000

3000

4000

5000

6000

7000

0 200 400 600 800 1000 1200

Use

r th

roug

hput

[kbp

s]


Mean cellMacroMicro



Figure 3.23: Mean user throughput in the network versus traffic demand per cell in the weightedinterference model.


3.7.5 Spatial distribution of QoS parameters averaged over many cellsin the network

In Section 3.7.3 we demonstrated the results for estimation of the QoS metrics as function ofthe traffic demand. Also, we assumed that all base stations transmit at the same power. In thissection we are interested in the spatial distribution of cell load, number of users and mean userthroughput considering that different base stations transmit at different powers. We consideronly the weighted interference case. Each BS v is characterized by a transmitting power Pv ∈ R∗+.We shall assume that Pv are i.i.d. (independent and identically distributed) marks of the pointprocess Φ (i.e., given Φ, the transmitting powers P1, P2, . . . are i.i.d. random variable with somefixed distribution).

The received power at location y ∈ R2 from BS u equals now

L−1u (y) =

PuSu (y − u)

` (y − u)(3.63)

The cell is now given by the follwing formula:

V (u) =

y ∈ R2 : Lu (y) ≤ min

v∈Φ\uLv (y)

(3.64)

The SINR formula given in (3.35) is now

SINR (y,Φ) :=PuSu (y − u) /l (|y − u|)

N +∑

v∈Φ\uPvSv (y − v) p (v) /l (|y − v|)

(3.65)

for a user at position y served by the base station u as a base station offering him the strongestsignal. Considering the different transmitting powers does not change the model itself developedin Section 3.5 and consequently we obtain the same system of cell-load equations as presentedin Section 3.5.3. Solving this system again we obtain the cell loads of all cells in a consideredarea (network). It is obvious from the results in Section 3.5.2 and Theorem 1 that solvingthe (3.36) one can obtain the cell load and further deduce the number of users and the meanuser throughput for all cells in the network. In such a way we obtain the cumulative distributionfunction of aforementioned parameters. The ultimate goal is to compare the results to thereal-field measurements.

Real-field measurements

Now we describe the real-field measurements. The raw data are collected using a specializedtool which is used by operational engineers for network maintenance. This tool measures severalparameters for every base station 24 hours a day. In particular, one can get the cell load, trafficdemand, number of users, mean user throughput for each cell in each hour. We have also theBS coordinates which permits to estimate the intensity γ of BS per unit area.

We choose one hour during the day and estimate the corresponding empirical CDF of theQoS parameters.

Numerical setup for simulation

The numerical setup is the same as in Section 3.7.3 except that the transmitting power is notconstant over the network. We assume that the transmitting power Pv has a log-normal distribu-tion of logarithmic-standard deviation σP . In order to justify this model, we give the empirical


CDF of transmitting powers in dB estimated from measurements in the operational network inFigure 3.24. This figure shows that this CDF may be approximated by a normal distributionwith standard deviation σP = 5.3dB. The mean transmitting power of each BS including a globalantenna gain equals E [Pn] = 60dBm.

40 45 50 55 60 65 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Total emitted power [dBm]

Cum

ulat

ive

dist

ribut

ion

func

tion

MeasuresNormal distribution

Figure 3.24: CDF of BS powers in the operational network in the downtown of a big city (blue)and normal distribution approximation (red).

With the constant power described in Section 3.5.1, each BS transmits at a constant powerPn = 60dBm and the shadowing (3.5.3) has a log-normal distribution of standard deviation

σS =√σ2S + σ2

P ' 9.6dB

The other sources of irregularities, as for example non-uniform traffic demand, are not con-sidered. Consequently, we will consider networks or parts of a network where we can assumeuniform spatial traffic demand (e.g. downtown of a big city or a typical rural area).

MEASURES mean standard deviationcell load 0.1854 0.1337mean number of users 0.2608 0.2511mean userthroughput[kbit/s] 2135 698SIMULATIONS mean standard deviationcell load 0.1845 0.1059mean number of users 0.2531 0.2147mean userthroughput[kbit/s] 2054 425

Table 3.1: Mean and standard deviation of spatial distribution of QoS metrics for the downtownof a big city

Results

Figures 3.25, 3.26 and 3.27 show the spatial distribution (across different cells) of the cellload, mean number of users per cell and the mean user throughput in the network deployed in


MEASURES mean standard deviationcell load 0.1190 0.1035mean number of users 0.1530 0.1734mean userthroughput[kbit/s] 1975 733SIMULATIONS mean standard deviationcell load 0.1321 0.0937mean number of users 0.1774 0.2861mean userthroughput[kbit/s] 2051 601

Table 3.2: Mean and standard deviation of spatial distribution of QoS metrics for the mid-sizecity

the downtown of a big city. Recall that these metrics represent, themselves, the steady-state(averaged over time) performance characteristics of individual cells.

0

0.2

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0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

Cum

ulat

ive

dist

ribut

ion

func

tion

Cell load

Variable powerMeasures

Constant power

Figure 3.25: CDF of cell load for the downtown of a big city obtained either from the variablepower model, from real-field measurements, or from the model where the transmitted powers areassumed constant.

Analogous characteristics regarding the network in a mid-size city are presented in Fig-ures 3.28, 3.29 and 3.30. Tables 3.1 and 3.2 show means and standard deviations of thesespatial distributions.

All figures and tables present the distributions estimates in our model as well as the real-fieldmeasurements. For sake of comparison, we present also in the figures the results obtained inthe model described in Section3.5.1 where the transmitted powers are assumed constant. Thesimulation curves represent the means over ten repeated network simulations, with the horizontalbars giving the standard deviation of this averaging. In what follows we discuss the presentedresults in more detail.

The estimated network density and the traffic demand in the downtown of the big city are,respectively, γ = 4.62km−2 and ρ = 483kbit/s/cell. Analogous values for the mid-size city areγ = 1.27km−2 and ρ = 284kbit/s/cell. Note that in the latter scenario the traffic demand issmaller, but the network is less dense and also less regular (cf Figure 3.16). We use these valuesas input parameters for our model.

In general we see a good agreement between real field measures and the model analysis with


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0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Cum

ulat

ive

dist

ribut

ion

func

tion

Number of users


Constant power

Figure 3.26: CDF of the mean users number for the downtown of a big city.

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000

Cum

ulat

ive

dist

ribut

ion

func

tion

User throughput [kbit/s]


Constant power

Figure 3.27: CDF of the throughput for the downtown of a big city.

0

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0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

Cum

ulat

ive

dist

ribut

ion

func

tion

Cell load


Constant power

Figure 3.28: CDF of cell load for a mid-size city obtained either from the variable power model,from real-field measurements, or from the model where the transmitted powers are assumedconstant.


0

0.2

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0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Cum

ulat

ive

dist

ribut

ion

func

tion

Number of users


Constant power

Figure 3.29: CDF of the mean number of users for the mid-size city.

0

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0.8

1

0 1000 2000 3000 4000 5000 6000

Cum

ulat

ive

dist

ribut

ion

func

tion

User throughput [kbit/s]


Constant power

Figure 3.30: CDF of the throughput for the mid-size city.

randomized transmitted power. Under the constant power assumption the model predicts well themedian of the cell load and the mean number of users but fails to match the spatial distribution ofthese characteristics. Clearly, the spatial variability of power creates more spatial heterogeneityof these characteristics in the network. Regarding the mean user throughput the constant powerassumption fails to predict even the median. Extensions of the model, e.g. letting it account forfurther sources of disparity in the deployed networks (e.g. different heights of antennas) couldperhaps improve the accuracy of prediction.

Appendix

3.A Proof of Proposition 4 in the Markovian case

Assume that the transmitted volumes are exponentially distributed. In this particular case, theprocess X (t) ; t ≥ 0 describing the number of users of the different classes is a continuous-timeMarkov process with discrete state space ND and admits the following generator

q (x, x+ εj) = λj , x ∈ ND

q (x, x− εj) = µjRjxjxD, x ∈ ND, xj > 0,

(3.66)

where εj designates the vector of ND having coordinate 1 at position j and 0 elsewhere andxD :=

∑j∈D xj the total number of users in the queue. It is easy to see that the process

X (t) ; t ≥ 0 is regular [27, p.337] and irreducible [27, p.357] and that it admits as invariantmeasure

α (x) = xD!∏j∈D

(ρ′j)xj

xj !, x ∈ ND, (3.67)

where ρ′j := λj/ (µjRj) = ρj/Rj . If ρ′ :=∑Jj=1 ρ

′j < 1 then

∑x∈ND α (x) = 1

1−ρ′ , indeed

1

1− ρ′=

∞∑n=0

ρ′n

=

∞∑n=0

∑j∈D

ρ′j

n

=

∞∑n=0

∑x∈ND:xD=n

n!∏j∈D

ρ′xjj

xj !

=∑x∈ND

xD!∏j∈D

(ρ′j)xj

xj !=∑x∈ND

α (x)

We deduce that if ρ′ < 1 then the process X (t) ; t ≥ 0 admits π = (1− ρ)α as invariantdistribution; and hence this process is t-positive recurrent [27, p.357]. We deduce from (3.67)that the invariant distribution is

π (x) = (1− ρ′)xD!∏j∈D

(ρ′j)xj

xj !, x ∈ ND

Let X = (X1, X2, . . . , XJ) be the vector counting the number of users of each class in the steadystate, and let XD :=

∑j∈DXj be the total number of users in the queue. The vector X has π as

93


distribution, then, for n ∈ N,

P (XD = n) =∑

x∈ND:xD=n

π (x)

= (1− ρ′)∑

x∈ND:xD=n

n!∏j∈D

(ρ′j)xj

xj !

= (1− ρ′) ρ′n,

which is the geometric distribution on N with parameter 1 − ρ′ = 1 − ρ/ρc where ρc is givenby (3.4). The mean number of users is

N := E [XD] =ρ′

1− ρ′=

ρ

ρc − ρ

From Little’s formula [11] the expected delay, denoted T , equals

T =E [XD]

λ=

ρ

(ρc − ρ)λ

In the steady state the queue throughput equals the traffic demand ρ. The throughput per useris defined as the ratio of the above queue throughput by the average number of users; that is

r =ρ

E [XD]= ρc − ρ

For a given class j ∈ D,

Nj := E [Xj ]

=∑x

xjπ (x)

=∑

x:xj 6=0

(1− ρ′)xD!xj∏i∈D

(ρ′i)xi

xi!

=∑x′

(1− ρ′) (x′D + 1)x′D!∏i∈D

(ρ′i)xi

x′i!

=∑x′

(x′D + 1)π (x′)

= ρ′jE [XD + 1] =ρj(

1− ρρc

)Rj

where for the fourth equality we introduce the vector x′ related to x as follows

x′i =

xi i 6= jxi − 1 i = j

From Little’s formula the expected delay, denoted Tj , equals

Tj =E [Xj ]

λj=

1(1− ρ

ρc

)Rjµj

3.A. PROOF OF PROPOSITION 4 IN THE MARKOVIAN CASE 95

The expected throughput of class j, denoted rj , is the average required volume µ−1j divided by

the expected delay, that is

rj =µ−1j

Tj=

(1− ρ

ρc

)Rj

Chapter 4

Quality of service in real-timestreaming

4.1 Introduction

Wireless cellular networks offer nowadays possibility to watch TV on mobile devices, whichis an example of the real-time content streaming. This type of traffic demand is expected toincrease significantly in the future. In order to cope with this process, network operators needto implement in their dimensioning tools efficient methods allowing to predict the quality of thistype of service. The quality of real-time streaming (RTS) is principally related to the numberand duration of outage incidents — (hopefully short) periods when the network cannot deliverto a given user in real-time the requested content of the required quality. In this Chapter wepropose a stochastic model allowing for an analytic evaluation of such metrics. It assumes a trafficdemand with different radio conditions of calls, and can be specified to take into account theparameters of a given wireless cellular technology. We develop expressions for several importantperformance characteristics of this model, including the mean time spent in outage and the meannumber of outage incidents for a typical streaming call as function of its radio conditions. Theseexpressions involve only stationary probabilities of the (free) traffic demand process, which is avector of independent Poisson random variables describing the number of users in different radioconditions.

We use this model to analyze RTS in a typical cell of a 3GPP Long Term Evolution (LTE)cellular network assuming orthogonal intra-cell user channels with the peak bit-rates (achievablewhen there are no other users in the same cell) close to the theoretical Shannon bound in theadditive white Gaussian noise (AWGN) channel, with the extra-cell interference treated as noise.These assumptions lead to a radio resource constraint in a multi-rate linear form. Namely,each user experiencing a given signal-to-(extra-cell)-interference-and-noise ratio (SINR) requiresa fixed fraction of the normalized radio capacity, related to the ratio between its requested andpeak bit-rates. All users of a given configuration (experiencing different SINR values) can beentirely satisfied if and only if the total required capacity is not larger than one.1

In the above context of a multi-rate linear radio resource constraint, we analyze some natural

1Recall that in the case of voice calls and, more generally, constant bit-rate (CBR) calls the multi-rate lin-ear form of the resource constraints has already proved to lead to efficient model evaluation methods, via e.g.Kaufman-Roberts algorithm [65, 81]. Despite some fundamental similarities to CBR service, the RTS gives riseto a new model, due to the fact that the service denials are not definitive for a given call, but have a form oftemporal interruptions (outage) periods.

97

98 CHAPTER 4. QUALITY OF SERVICE IN REAL-TIME STREAMING

parametric class of least-effort-served-first (LESF) service policies, which assign service to usersin order of their increasing radio capacity demand, until the full capacity (possibly with somemargin) is reached. The capacity margin may be used to offer some “lower quality” service tousers temporarily in outage thus realizing some type of fairness with respect to unequal userradio-channel conditions. This class contains an optimal and a fair policy, the latter beingsuggested by LTE implementations.

In order to evaluate explicitly the quality of service metrics induced by the LESF policies,we relate the mean time spent in outage and the mean number of outage incidents for a typicalstreaming call in given radio conditions to the distribution functions of some linear functionals ofthe Poisson vector describing the steady state of the system. We calculate the Fourier transformsof these functions and use a well-known Fourier transform inversion method to obtain numericalvalues of the quantities of interest. We also study the mean throughput during a typical streamingcall evaluating the expectations of the corresponding non-linear functionals of the Poisson vectordescribing the steady state of the system via the Monte Carlo method.

Using this approach, we present a thorough study of the quality of RTS with LESF policies inthe aforementioned Markovian setting. For completeness we present also some pure-simulationresults illustrating the impact of a non Poisson-arrival assumption.

4.2 Related work

Let us now recollect a few related works on the performance evaluation of cellular networks.In the early 80’s, wireless cellular networks were carrying essentially voice calls, which requireconstant bit-rates (CBR) and are subject to admission control policies with blocking (at thearrival epoch) to guarantee these rates for calls already in service. An important amount of workhas been done to propose efficient call admission policies [83, 100, 102]. Policies with admissionconditions in the multi-rate linear form have been considered e.g. in [10,42,59].

Progressively, cellular networks started carrying also calls with variable bit-rates (VBR), usedto transmit data files. The available resources are (fairly) shared between such calls and when thetraffic demand increases, the file transfer delays increase as well, but (in principle) no call is everblocked. These delays may be evaluated analytically using multi-rate linear resource constraintin conjunction with multi-class processor sharing models; cf e.g. [25, 59].

Recently, users may access multimedia streaming services through their mobile devices [45].They are provided via CBR connections, essentially without admission control, but they toleratetemporary interruptions, when network congestion occurs. One may distinguish two types ofstreaming traffic. In real-time streaming (RTS) (as e.g. in mobile TV), considered in this thesis,the portions of the streaming content emitted during the time when the transmission to a givenuser is interrupted (is in outage) are definitely lost for him (unless a “secondary”, lower-ratestreaming is provided during these periods). In non-real-time streaming (NRTS) (like e.g., video-on-demand, YouTube, Dailymotion, etc), a user starts playing back the requested multimediacontent after some initial delay, required to deliver and buffer on the user device some initialportion of it. If further transmission is interrupted for some time making the user buffer contentdrop to zero (buffer starvation) then the play-back is stopped until some new required portion ofthe content is delivered. Several papers study the effect of the variability of the wireless channelon the performance of a single streaming call; see for e.g. [68], [75]. In [82] VBR transmissions andRTS are considered jointly in some analytical model, however the number and duration of outageperiods are not evaluated. In [99] the tradeoff between the start-up delay and the probability ofbuffer starvation is analyzed in a Markovian queuing framework for NRTS streaming.

We do not consider any cell-load balancing; see [14] for some recent work on this problem in

4.3. STREAMING IN WIRELESS CELLULAR NETWORKS 99

the video streaming context. Also, [67,98] consider some admission control policies to guaranteenon-dropping of multimedia calls due to caller impatience and/or handoffs.

4.3 Streaming in wireless cellular networks

In this section we present a new stochastic model of RTS in cellular networks.

4.3.1 System assumptions

We consider the following scenario of multi-user streaming in a cellular network.

Network layer

Geographically distributed users wish to obtain down-link wireless streaming of some (typicallyvideo) content, contacting base stations of a network at random times, for random durations,requesting some fixed streaming bit-rates. We consider a uni-cast traffic (as opposed to thebroadcast or multi-cast case), i.e.; the content is delivered to all users via private connections.Different classes of users (calls) need to be distinguished, regarding their radio channel conditions,requested streaming bit-rates and mean streaming times. Each user chooses one base station,the one with the smallest path-loss, independently of the configuration of users served by thisbase station. Thus, we do not consider any load-balancing policy.

Data layer — streaming policies

If a given base station cannot serve all the users present at a given time, it temporarily stopsstreaming the requested content at the requested rate to users of some classes, according tosome given policy (to be described), which is supposed to preserve a maximal subset of servedusers. We call these (classes of) users with the requested bit-rate temporarily denied in outage.The users in outage will not receive the part of the content which is transmitted during theiroutage times (this is the principle of the RTS). We will also consider policies, which offer some“best-effort” streaming bit-rates for some classes of users in outage, thus allowing for exampleto keep receiving the requested content but of a lower quality. Users, which are (temporarily)denied even this lower quality of service are called in deep outage.

Medium access

In this thesis we assume that users are connected to the serving antennas via orthogonal single-input-single-output (SISO) channels allowing for a peak-rate close to the theoretical Shannonbound in the additive white Gaussian noise (AWGN) model, with the (extra-cell) interfer-ence treated as noise.2 We will also comment on how to model multiple-input-multiple-output(MIMO) and broadcast channels.

Physical layer

The quality of channel of a given user depends on the path-loss of the signal with respect toits serving base station, a constant noise, and the interference from other (non-serving) basestations. These three components determine its signal-to-interference-and-noise ratio (SINR).

2Orthogonality of channels is an appropriate assumption for current LTE (Long Term Evolution) norm forcellular networks based on OFDMA, as well as for other multiple access techniques as FDA, TDMA, CDMAassuming perfect in-cell orthogonality, and even HDR neglecting the scheduler gain.


Both path-loss from the serving station and interference account for the distance and randompropagation effects (shadowing). Our main motivation for considering a multi-class model is todistinguish users with different SINR values. In other words, even if we assume that all usersrequire the same streaming times and rates, we still need a multi-class model due to (typically)different SINR’s values of users in wireless cellular networks.

Performance characteristics

We will present and analytically evaluate performance of some (realistic) streaming policies inthe context described above. We will be particularly interested in the following characteristics:

• fraction of time spent in outage and in deep outage during the typical call of a given class,

• number of outage incidents occurring during this call,

• mean throughput (average bit-rate) during such call, accounting for the requested bit-ratesand for the “best-effort” bit-rate obtained during the outage periods.

4.3.2 Model description

In what follows we describe a mathematical model of the RTS that is an incarnation of a new,more general, stochastic service model with capacity sharing and interruptions presented andanalyzed in Appendix 4.4.2. This is a single server model which allows to study the performanceof one tagged base station of a multi-cellular network satisfying the above system assumptions.More details on how this model fits the multi-cell scenario will be presented in Section 4.4.

Traffic demand

Consider J ≥ 1 classes of calls (or, equivalently, users) characterized by different requestedstreaming bit-rates rk, wireless channel conditions described by the signal-to-(extra-cell)-interference-and-noise ratio SINRk with respect to the serving base-station 3 and mean requested streamingtimes 1/µk, k = 1, . . . , J .

We assume that calls of class k ∈ 1, . . . , J arrive in time according to a Poisson process withintensity λk > 0 (number of call arrivals per unit of time, per base station) and stay in the system(keep requesting streaming) for independent times, having some general distribution with mean1/µk <∞. 4 Different classes of calls are independent from each other. We denote by Xk(t) thenumber of calls of a given class requesting streaming from a given BS at time t; see Section 4.1.1in the Appendix for a formal definitions of these variables in terms of arrival process and servicetimes. Let X(t) = (X1(t), . . . , XJ(t)); we call it the (vector of) user configuration at time t. Thestationary distribution π of X(t) coincides with the distribution of the vector (X1, . . . , XJ) ofindependent Poisson random variables with means E[Xk] := ρk = λk/µk, k = 1, 2, . . . , J . Wecall ρk the traffic demand (per base station) of class k.

Wireless resource constraints

Users are supposed to be offered the requested streaming rates for the whole requested streamingtimes. However, due to limited wireless resources, for some configuration of users X(t), therequested streaming rates r = (r1, . . . , rJ) may not be achievable. Following the assumption of

3In this thesis the interference is always caused only by non-serving base stations.4All the results presented in this Chapter do not depend on the particular choice of the streaming time

distributions. This property is often referred to in the queuing context as the insensitivity property.


orthogonal AWGN SISO wireless channels (with the (extra-cell) interference treated as noise)available for users of a given base station, we assume that the requested rates are achievable forall calls present at time t if

Xk(t)rk = νkRk, k = 1, . . . , J, (4.1)

for some non-negative vector (ν1, . . . , νJ), such that∑Jk=1 νk ≤ 1, where

Rk = cW log(1 + SINRk) (4.2)

is the maximal (peak) bit-rate of a user of class k, whose channel conditions are characterizedby SINRk. (The rate Rk is available to a user of class k if it is the only user served by the basestation.) Here W is the frequency bandwidth and c (with 0 < c ≤ 1) is a coefficient capturinghow close a given coding scheme approaches the theoretical Shannon bound (corresponding toc = 1); cf [34, Th .9.1.1]. 5 Note that the assumption (4.1) corresponds to the situation, whenusers neither hamper nor assist each other’s transmission. They use channels which are perfectlyseparated in time, frequency or by orthogonal codes, nevertheless sharing these resources. 6

We can interpret the ratio between the requested and maximal bit-rates ϕk = rk/Rk as theresource demand of a user of class k. Note that the configuration of users X(t) can be entirelyserved if and only if the total resource demand satisfies the constraint

J∑k=1

ϕkXk(t) ≤ 1 . (4.3)

This is a multi-rate linear resource constraint.

Service policy

If the requested streaming rates are not achievable for a given configuration of users X(t) presentat time t, then some classes of users will be temporarily put in outage at time t, meaning thatthey will receive some smaller bit-rates (whose values are not guaranteed and may depend onthe configuration X(t)). These smaller, “best-effort” bit-rates may drop to 0, in which case wesay that users are in deep-outage. Let us recall that the times at which users are in outage anddeep outage do not alter the original streaming times; i.e. the streaming content is not buffered,nor delayed during the outage periods.

We will define now a parametric family of service polices for which classes with smallerresource demands have higher service priority. In this regard, in the remaining part of thisSection we assume (without loss of generality) that the resource demands of users from differentclasses are ordered ϕ1 < ϕ2 < . . . < ϕJ .

5It was also shown in [61] that the performance of AWGN multiple input multiple output (MIMO) channel canbe approximated by taking values of γ ≥ 1. Another possibility to consider MIMO channel is to use the exactcapacity formula given in [90].

6From an information theory point of view, the orthogonality assumption is not optimal. In fact, the theo-retically optimal performance is offered by the broadcast channel model. It is known that in the case of AWGNbroadcast channel the rates r are (theoretically) achievable for the configuration X if (and only if) there exists a

vector (ν1, . . . , νJ ), such that∑J

k=1 νk ≤ 1 and

Xkrk = W log

(1 +

νk

1/SINRk +∑k−1

i=1 νi

)k = 1, . . . , J,

where the classes of users are numbered such that SINR1 ≥ SINR2 ≥ . . . ≥ SINRJ ; cf [92, Eq. 6.29].


Least-effort-served-first policy For a given configuration of users X = X(t) requestingstreaming at time t, least-effort-served-first policy with δ-margin (LESF(δ) for short) attributesthe requested bit-rates to all users in classes k = 1, . . . ,K, where

K =Kδ(X) = max k ∈ 1, . . . , J : (4.4)

k−1∑j=1

ϕjXj + ϕk

J∑j=k

Xj1(ϕj ≤ ϕk(1 + δ)) ≤ 1

,

where 1A(x) = 1 is the indicator function of set A and δ is a constant satisfying 0 ≤ δ ≤ ∞.

Remark 27 The LESF(0) policy is optimal in the following sense: given constraint (4.3) andthe assumption that the classes with smaller resource demands have higher priority, this policyallows to serve the maximal subset of users present in the system. For the same reason anyLESF(δ) policy with δ > 0 is clearly sub-optimal. In order to explain the motivation for consid-ering such policies, one needs to extend the model and explain what actually happens with classesof users which experience outage. In this regard, note that C =

∑Kj=1 ϕjXj ≤ 1 is the actual

fraction of the server capacity consumed by the users which are not in outage. The remainingserver capacity 1 − C (which is not needed to serve users in classes 1, . . . ,K) can be used tooffer some “lower quality” service (e.g. streaming with lower video resolution, etc) to the usersin classes K + 1, . . . , J which are in outage. Note by (4.4) that the remaining server capacityunder the policy LESF(δ) is at least

1− C ≥ ϕKJ∑

j=K+1

Xj1(ϕj ≤ ϕK(1 + δ)) .

Hence, the server accepting the class K as the least-priority class being “fully” served, leavesenough remaining capacity to be able to make the same effort (allocate service capacity ϕK) forall users in outage in classes whose service demand exceeds ϕK by no more than δ × 100%.These latter users will not have “full” required service (since this requires more resources, ϕj >ϕK , for the full service) but only some “lower quality” service (to be specified in what follows).Consequently, one can conclude that policies LESF(δ) with δ > 0, being sub-optimal, ensure somefairness, in the sense explained above. Clearly the policy LESF(∞) (i.e., with δ =∞) is the mostfair, in the sense that it reserves enough remaining capacity to offer the “lower quality” servicefor all users in outage (no deep outage). Thus, we will call LESF(∞) the LESF fair policy.

Best-effort service for users in outage We will specify now a natural model for the “best-effort” streaming bit-rates that can be offered for users in outage in association with a givenLESF(δ) policy. For k > K = Kδ(X) denote

r′k = r′δk (X) = Rk

1−∑Kj=1Xjϕj∑J

j=K+1Xj1(ϕj ≤ (1 + δ)ϕK)(4.5)

if ϕk ≤ (1 + δ)ϕK and 0 otherwise.

The rates (r1, . . . , rK , r′K+1, . . . , r

′J) are achievable for the configuration X under resource con-

straint (4.3). Note that users in classes j such that ϕj > (1 + δ)ϕK do not receive any positivebit-rate. We say, they are in deep outage. Finally, we remark that the service (4.5) is “resourcefair” among users in outage but not in deep outage.


Performance metrics

The configuration of users X(t) evolves in time, it changes at arrival and departure times ofusers. At each arrival or departure epoch the base station applies the outage policy to the newconfiguration of users to decide which classes of users receive requested streaming rates and whichare in outage (or deep outage).

Let us introduce the following characteristics of the typical call (user) of class k = 1, . . . , J .

• Pk denotes the probability of outage at the arrival epoch for class k. This is the probabilitythat the typical call of this class is put in outage immediately at its arrival epoch.

• Dk denotes the mean total time spent in outage during the typical call of class k.

• Mk denotes the mean number of outage incidents experienced during the typical call of classk.

More formal definitions of these characteristics, as well as other system characteristics (ase.g. the intensity of outage incidents) are given in the Appendix. We also introduce two furthercharacteristics related to the mean throughput obtained during the typical call of class k =1, . . . , J .

• Denote by Υk the mean throughput during the typical call of class k. This is the meanbit-rate obtained during such a call, taking into account the bit-rate rk when the call isnot in outage and the best-effort bit rate r′k obtained during the outage periods, averagedover the call duration.

• Let Υ′k be the part of the throughput obtained during the outage periods of the typical callof class k. This is the mean best-effort bit-rate of such call averaged over outage periods.

4.3.3 Model evaluation

Results

We will show how the performance metrics regarding outage incidents and duration, introduced inSection 4.3.2, can be expressed using probability distribution functions of some linear functionalsof the random vector X1, . . . , XJ of independent Poisson random variables with parameters ρj ,respectively. Recall that these random variables correspond to the number of calls of differentclasses present in the stationary regime of our streaming model.

Specifically, for given δ > 0, k = 1, . . . , J and t ≥ 0 denote

F δk (t) := Pr

k∑j=1

Xδ,kj ϕj ≤ t

, (4.6)

where Xδ,kj = Xj for j = 1, . . . , k − 1 and Xδ,k

k =∑Jj=kXj1(ϕj ≤ ϕk(1 + δ)).

The following results follow from the analysis of a more general model presented in theAppendix.

Proposition 11 The probability of outage at the arrival epoch for user of class k is equal to

Pk = 1− F δk (1− ϕk) k = 1, . . . , J . (4.7)


The mean total time spent in outage during the typical call of class k is equal to

Dk =Pkµk

=1− F δk (1− ϕk)

µkk = 1, . . . , J . (4.8)

The mean number of outage incidents experienced during the typical call of class k (after itsarrival) is equal to

Mk =1

µk

J∑j=1

λj(F δk (1− ϕk)− F δk (1− ϕk − ϕj)

)k = 1, . . . , J . (4.9)

Proof. Note first that the functions F δk (t) defined in (4.6) allow one to represent the station-ary probability that the configuration of users is in a state in which the LESF(δ) policy servesusers of class k

F δk (1) = Pr

k∑j=1

Xδ,kj ϕj ≤ 1

.

In the general model described in the Appendix we denote this state by Fk and its probabilityby π(Fk). Thus π(Fk) = F δk (1). Moreover,

1− F δk (1− ϕk) = Pr

k∑j=1

Xδ,kj ϕj > 1− ϕk

is the probability that the steady state configuration of users appended with one user of class k isin the complement F ′k of the state Fk, i.e., all users of class k are in outage (meaning k > Kδ(X′),where X′ = (X1, . . . , Xk + 1, . . . , XJ)). Thus the expression (4.7) follows from Proposition 13.Similarly (4.8) follows from Proposition 14 and (4.9) follows from Proposition 15.

Regarding the throughput characteristics, we have the following result.

Proposition 12 The mean throughput during the typical call of class k is equal to

Υk = rk(1− Pk) + Υ′k = rkFδk (1− ϕk) + Υ′k ,

where

Υ′k = E[r′δk (X1, . . . , Xk + 1, . . . , XJ) (4.10)

1(Kδ(X1, . . . , Xk + 1, . . . , XJ) < k

)],

with the best-effort rate r′k(·) given by (4.5) and the least-priority class Kδ(·) begin served by theLESF(δ) policy given by (4.4), is the part of the throughput obtained during the outage periods.

Proof of this proposition is given in the Appendix.

Remark 28 Recall from (4.5) that the variable rates r′k are obtained by the user of class k whenhe is in outage, i.e., k > K. They are non-null, r′k > 0, only if ϕk ≤ (1 + δ)ϕK . In the case ofequal requested rates rk, the intersection of the two conditions 0 < r′k and k > K is equivalent to

(1 + SINRK)1/(1+δ) − 1 ≤ SINRk ≤ SINRK . (4.11)


Remarks on numerical evaluation

In order to be able to use the expressions given in (11) we need to evaluate the distributionfunctions F δk (t). In what follows we show how this can be done using Laplace transforms.Regarding the throughput in outage Υ′k, expressed in (4.10) as the expectation of a non-linearfunctional of the vector (X1, . . . , XJ), we will use Monte Carlo simulations to obtain numericalvalues for this expectation.

Denote by Lδk(θ) :=∫∞

0e−θsF δk (s)ds the Laplace transform of the function F δk (t).

Fact 2 We have

Lδk(θ) =1

θexp

k∑j=1

ρδ,kj(e−θϕj − 1

) ,where ρδ,kj = ρj for j = 1, . . . , k − 1 and ρδ,kk =

∑Jj=k ρj1(ϕj ≤ ϕk(1 + δ)).

Proof. Note that for given δ > 0, k = 1, . . . , J the random variables Xδ,k1 , . . . , Xδ,k

k are

independent, of Poisson distribution, with parameters ρδ,k1 , . . . , ρδ,kk , respectively. The resultfollows from [9, Proposition 1.2.2] and a general relation

∫∞0e−θsF (s) ds = 1

θ

∫∞0e−θsF (ds).

The probabilities F δk (·) may be retrieved from Lδk(·) using standard techniques. For ex-ample with the algorithm implemented by [55] in Matlab [38]. In what follows we present amore explicit result based on the Bromwich contour inversion integral. In this regard, denote

Lδ

k(θ) = 1/θ − Lδk(θ) (which is the Laplace transform of complementary distribution function1− F δk (t)). Also, denote by R(z) the real part of the complex number z.

Fact 3 We have

F δk (t) = 1− 2eat

π

∫ ∞0

R(Lδ

k(a+ iu)

)cosut du , (4.12)

where a > 0 is an arbitrary constant.

Proof. See [6].

Remark 29 As shown in [6], the integral in (4.12) can be numerically evaluated using thetrapezoidal rule, with the parameter a allowing to control the approximation error. Specifically,for n = 0, 1, . . . define

hn(t) = hn(t; a, k, δ) :=(−1)nea/2

tR(Lδ

k

(a+ 2nπi

2t

)),

Sn(t) := h0(t)2 +

∑ni=1 hi(t), and S(t) = limn→∞ Sn(t). Then

∣∣F δk (t)− (1− S(t))∣∣ ≤ e−a. Finally,

the (alternating) infinite series S(t) can be efficiently approximated using for example the Eulersummation rule

S(t) ≈M∑i=0

(M

i

)2−MSN+i(t)

with a typical choice N = 15, M = 11.

Remark 30 The expression (4.9) for the mean number of outage incidents involves a sum of apotentially big number of terms F δk (1− ϕk)− F δk (1− ϕk − ϕj), j = 1, . . . , J , which are typicallysmall, and which are evaluated via the inversion of the Laplace transform. Consequently the sum


may accumulate precision errors. In order to avoid this problem we propose another numericalapproach for calculating Mk. It consists in representing Mk equivalently to (4.9) as

Mk =F δk (1− ϕk)

µk

J∑j=1

λjbk(j) k = 1, . . . , J (4.13)

where

bk (j) =F δk (1− ϕk)− F δk (1− ϕk − ϕj)

F δk (1− ϕk)(4.14)

Let k and δ be fixed. Recall the definition of F δk (t) in (4.6) and note that the expression (4.14)may be written as

bk (j) =Pr (X ∈ F , X + εj /∈ F)

Pr (X ∈ F)

where F = F(k) =X ∈ RJ :

∑kj=1X

δ,kj ϕj ≤ 1− ϕk

. The above expression may be seen as

the blocking probability for class j in a classical multi-class Erlang loss system with the admis-sion condition X ∈ F . Consequently, bk (·) may be calculated by using the Kaufman-Robertsalgorithm [65, 81] and plugged into (4.13). Note that by doing this we still need to calculateF δk (1− ϕk) however avoid summing of O differences of these functions as in (4.9).

4.4 Quality of real-time streaming in LTE

In this section we will use the model developed in Section 4.3 to evaluate the quality of RTS inLTE symmetric networks. This single-server (base station) model will be used to study the per-formance of one tagged base station of a multi-cellular network under the following assumptions:

• We assume a regular hexagonal lattice of base stations on a torus. This allows us to considerthe tagged base station of the network as a typical one.

• Homogeneous (in space and time) Poisson arrivals on the torus are marked by i.i.d. (acrossusers and base stations) variables representing their shadowing with respect to differentbase stations. These variables, together with independent user locations determine theirserving (strongest) base stations. A consequence of the independence of users locationsand shadowing variables is that the arrivals served by the tagged base station form anindependent thinning of the total Poisson arrival process to the torus and thus a Poissonprocess too. Uniform distribution of user locations and identical distribution of the theirshadowing variables imply that the intensity of the arrival process to the tagged basestation is equal to the total arrival intensity to the torus divided by the number of stations.Moreover, the distribution of the SINR of the typical user of the tagged base stationcoincides with the distribution of the typical user of the whole network.

• The intensity of arrivals of some particular (SINR)-class to the tagged base station is equalto the total intensity of arrivals to the tagged cell times the probability of the randomSINR of the typical user being in the SINR-interval corresponding to this class.

• We consider the “full interference” scenario, i.e. all base stations transmit the signal at theconstant power, regardless of the number of users they serve (this number can be zero).This makes the interference, and hence the service rates, of users of a given base stationindependent of the service of other base stations (decouples the service processes of differentbase stations).

4.4. QUALITY OF REAL-TIME STREAMING IN LTE 107

4.4.1 LTE model and traffic specification

SINR distribution

Recall that the main motivation for considering a multi-class model was the necessity to dis-tinguish users with different radio conditions, related to different values of the SINR they havewith respect to the serving base stations. In order to choose representative values of SINR in agiven network and to know what fraction of users experience a given value, we need to know the(spatial) distribution of the SINR (with respect to the serving base station) experienced in thisnetwork (possibly biased by the spatial repartition of arrivals of streaming calls). This distribu-tion can be obtained from real-network measurements, simulations or analytic evaluation of anappropriate spatial, stochastic model.7 In this Section we will use the distribution of SINR ob-tained from the simulation compliant with the 3GPP recommendation in the so-called calibrationcase (to be explained in what follows). At present, assume simply, that we are given a cumula-tive distribution function (CDF) of the SINR expressed in dB, F (x) := Pr10 log10(SINR) ≤ x,obtained from either of these methods. In other words, F (x) represents the fraction of mobileusers in the given network which experience the SINR (expressed in dB) not larger than x.

Consider a discrete probability mass function

pk := F

(xk+1 + xk

2

)− F

(xk + xk−1

2

)k = 1, 2, . . . , J , (4.15)

with x0 = −∞, xJ+1 = ∞. We define the class k = 1, . . . , J of users as all users having

the SINR expressed in dB in the interval(

(xk+xk−1)2 , (xk+1+xk)

2

), and approximate their SINR

by the common value SINRk = 10xk/10. Clearly pk is the fraction of mobile users in the givennetwork which experience the SINR close to SINRk. Hence, in the case of homogeneous streamingtraffic (the same requested streaming rates and mean streaming times, which will be our defaultassumption in the numerical examples) we can assume the intensity of arrivals λk of users of

class k to be equal to λk = pkλ where λ =∑Ji=k λk is the total arrival intensity (per unit of time

per serving base station) to be specified together with the CDF F of the SINR.

CDF of the SINR for 3GPP recommendation We obtain the CDF F of the SINR from thesimulation compliant with the 3GPP recommendation in the so-called calibration case, (compareto [3]). More precisely, we consider the geometric pattern of BS placed on the 6 × 6 hexagonallattice. In the middle of each hexagon there are three symmetrically oriented BS antennas,which gives a total of 108 BS antennas. The distance between the centers of two neighboringhexagons is 0.5 km. Each BS antenna is characterized by the following horizontal pattern A(φ) =−min(12(φ/ζ)2, Am), where φ is the angle in degrees, with ζ = 70, Am = 20dB, and usestransmission power P = 60dBm (including omnidirectional gain of 14dBi). The distance-lossmodel (corresponding to the frequency carrier 2GHz) is L(r) = 128.1 + 37.6× log10(r)[dB] wherer is the distance in km. A supplementary penetration loss of 20dB is added. The shadowingis modeled as a log-normal random variable of mean one and logarithmic standard deviationof deviation 8dB, cf [18]. The noise power equals −95dBm (which corresponds to a systembandwidth of 10MHz, a noise floor of −174dBm/Hz and a noise figure of 9dB). In order toobtain the empirical CDF of the SINR we generate 3600 random user locations uniformly in thenetwork (100 user locations per hexagon on average). Each user is connected to the antennawith the strongest received signal (smallest propagation-loss including distance, shadowing and

7For this latter possibility, we refer the reader to a recent paper on Poisson modeling of real cellular networkssubject to shadowing [20], as well as to [39], completed in [19], where the distribution of the the SINR in Poissonnetworks is evaluated explicitly.


antenna pattern) and the SINR is calculated. The obtained empirical CDF F of the SINR isshown in Figure 4.1.

Figure 4.1: Cumulative distribution function of the SINR obtained according to 3GPP specifica-tion; see Section 4.4.1. An abrupt transition of the CDF to 1 at SINR = 17dB is due to the cellsectorization: each mobile is interfered by each of the two antennas co-located with its servingantenna on the same site (and serving the different sectors) with the power equal to at least 1%of the power received from the serving BS. Therefore the signal to interference ratio is at most0.5× 10−2 = 17dB.

Link characteristics

3GPP shows in [4] that there is a 25% gap between the practical coding schemes and the Shannonlimit for the AWGN channel. Moreover, some of the transmitted bits are used for signaling,which induces a supplementary capacity loss of about 30% (see [2]). This made us assumec = 0.5(≈ 0.75(1− 0.3)) in (4.2). The system bandwidth is W = 10MHz.

Streaming traffic

We assume that all calls require the same streaming rate rk = 256 kbit/s and have the samestreaming call time distribution. We split them into J = 100 user classes characterized byvalues of the SINR falling into different intervals regularly approximating the SINR domainfrom x1 = −10dB to xJ = 17dB as explained in Section 4.4.1. In our performance evaluationwe will consider two values of the spatially uniform traffic demand: 900 and 600 Erlang/km2.(All results presented in what follows do not depend on the mean streaming time but only on


the traffic demand). Consequently, the k th class traffic demand per unit of area is equal to,respectively, pk × 900 and pk × 600 Erlang/km2, where pk are given by (4.15). Multiplyingby the area served by one base station equal to

√3 · (0.5 km)2/6 ≈ 0.0722 km2 we obtain the

traffic demand per cell, per class, equal to ρk = pk × 900 × 0.0722 ≈ pk × 64.9 Erlang andρk = pk × 600× 0.0722 ≈ pk × 43.3 Erlang, respectively, for the two studied scenarios.

4.4.2 Performance evaluation

Assuming the LTE and traffic model described above, we consider now streaming policies LESF(δ)defined in Section 4.3.2. Recall that in doing so, we assume that users are served by the antennaoffering the smallest path-loss, and dispose orthogonal down-link channels, with the maximalrates Rk depending on the value of the SINR (interference comes from non-serving BS) charac-terizing class k. Roughly speaking, LESF(δ) policy assigns the total requested streaming raterk = 256kbit/s for the maximal possible subset of classes in the order of decreasing SINR, leavingsome capacity margin to offer some “best-effort” streaming rates for (some) users remaining inoutage. These streaming rates r′k given by (4.5) depend on the current configuration of users andare non-zero for users with SINR within the interval (1 + SINRK)1/(1+δ) − 1 ≤ SINR ≤ SINRK ,where SINRK is the minimal value of SINR for which users are assigned the total requestedstreaming rate; cf Remark 28. In particular, LESF(0), called the optimal policy, leaves no ca-pacity margin for users in outage, while LESF(∞), called the fair one, offers a “best-effort”streaming rate for all users in outage at the price of assigning the full requested rate 256kbit/sto a smaller number of classes (higher value of the SINRK) 8. In what follows, we use our resultsof Section 4.3.3 to evaluate performance of these streaming policies in the LTE network model.

Outage time

Figure 4.2 shows the mean time of the streaming call spent in outage normalized by call duration,µkDk, evaluated using (4.8), as function of the SINR value characterizing class k, for the traffic900 Erlang/km2 and different policies LESF(δ). Figure 4.3 shows the analogous results assuminga traffic load of 600 Erlang/km2. The main observations are as follows:

• All LESF policies exhibit a cut-off behaviour: the fraction of time in outage drops rapidlyfrom 100% to 0% when the SINR exceeds some critical values. This cut-off is more strictfor the optimal policy.

• For the traffic of 900 Erlang/km2, users with the SINR≥ 3dB are practically never inoutage, when the optimal policy is used. The same holds true for users with SINR≥ 13dB,when the fair policy is used.

• When the traffic drops to 600 Erlang/km2, these critical values of SINR decrease by 2dBand 5dB, respectively, for the optimal and the fair policy. Note that the fair policy is moresensitive to higher traffic load.

Number of outage incidents

Figure 4.4 shows the mean number of outage incidents per streaming call, Mk evaluated us-ing (4.9), as function of the SINR value characterizing class k, for the traffic 900 Erlang/km2

and different policies LESF(δ). (Recall that we assume the same streaming time distributionfor all users, and hence λj/µk = ρj making the expression in (4.9) depend only on the vector of

8The LESF fair policy seems to be adopted in some implementations of the LTE.


Figure 4.2: Mean fraction of the requested streaming time in outage, as function of the userSINR for different policies LESF(δ); traffic 900 Erlang/km2.

traffic demand per class.) Figure 4.5 shows the analogous results assuming a traffic demand of600 Erlang/km2. The main observations are as follows:

• For all policies, the number of outage incidents (during the service) is non-zero only forusers with the SINR close to the critical values revealed by the analysis of the outage times.Users with SINR below these values are constantly in outage while users with SINR abovethem are never in outage.

• More fair policies generate slightly more outage incidents. The worst values are 2 to 2.2interruptions per call for the optimal policy, depending on the traffic value, and 2.4 to 3interruptions per call for the fair policy.

Studying outage times and outage incidents we do not see apparent reasons for consideringfair policies. This motivates our study of the best-effort service in outage.

The role of the ”best effort” service

Figure 4.6 shows the fraction of time spent in deep outage as function of the SINR, assumingtraffic 900 Erlang/km2. These values should be compared to the fraction of time spent in outage(for convenience copied in Figure 4.6 from Figure 4.2). Recall, users in outage do not receivethe full requested streaming rate (assumed 256kbit/s in our example), however they do receivesome non-null “best effort” rates given by (4.5), unless they are in deep outage — have SINR toosmall; cf Remark 28. Considering users in outage but not in deep outage as “partially satisfied”,


Figure 4.3: Fraction of time in outage as on Figure 4.2 for traffic 600 Erlang/km2.

increasing fairness margin δ allows to (at least) partially satisfy users with decreasing SINRvalues. Obviously the level of the “partial satisfaction” depends on the throughput obtained inoutage periods, which is our quantity of interest in Figure 4.7. It shows also two curves for allpolicies LESF(δ) assuming traffic 900 Erlang/km2. The upper ones represent the mean totalthroughput realized during the service, normalized to its maximal value; i.e., Υk/(256kbit/s),in function of the SINR value characterizing class k. The fractions of this throughput realizedduring outage periods, Υ′k/(256kbit/s), are represented by the lower curves.

Figures 4.7 and 4.6 teach us that the role of the LESF(δ) policies with δ > 0 may be two-fold.

• LESF(δ) policies with small values of δ, e.g. δ = 0.5, improve “temporal homogeneity” ofservice with respect to the optimal policy, for users having SINR near the critical value.For example, a user having SINR equal to 1dB is served by the optimal policy during80% of the time with the full requested streaming rate (cf. Figure 4.6). However, for theremaining 20% of the time it does not receive any service (deep outage, rate 0bits/s). Thepolicy LESF(0.5) offers to such a user 80% of the requested streaming rate during the wholestreaming time (cf. Figure 4.7), with no deep outage periods (cf. Figure 4.6). The pricefor this is that a slightly higher SINR is required to receive the full requested streamingrate (at least 5dB, instead of 3dB for the optimal policy).

• The fair policy LESF(∞) improves the spatial homogeneity of service. It leaves no userin deep outage, however a much larger SINR= 13dB is required for not to be in outage(cf. Figure 4.6). Moreover, the throughput of all users in outage but not in deep outageis substantially reduced e.g. from 80% to 40% for SINR= 1dB, with respect to some


Figure 4.4: Number of outage incidents during the requested streaming time, as function of theuser SINR for different policies LESF(δ); traffic 900 Erlang/km2.

intermediate LESF(δ) policies (with 0 < δ <∞). These intermediate policies can offer aninteresting compromise between the optimality and fairness.

Impact of non-Poisson-arrivals

Recall that the performance analysis of the model presented in this Chapter is insensitive to thedistribution of the requested streaming times. In this section we will briefly study the impact ofa non Poisson-arrival assumption. In this regard we simulate the dynamics of the model withdeterministic inter-arrival times (with all other model assumptions as before) and estimate themean fraction of time in outage µkDk and mean number of outage incidents Mk for each classk. For the comparison, as well as for the validation of the theoretical work, we perform also thesimulation of the model with Poisson arrivals. The results are plotted in Figures 4.8, 4.9 and4.10, 4.11. Observe first that the simulations of the Poisson model confirm the results of thetheoretical analysis. Regarding the impact of the deterministic inter-arrival times a (somewhatexpected) fact is that the optimal policy remains optimal regarding the fraction of time spentin the outage and the number of outage incidents. Another, less evident, observation is that thedeterministic inter-arrivals (more regular than in the Poisson case) do not improve the situationfor all classes of users. In fact, users with small values of the SINR have a smaller fraction oftime in outage under Poisson arrival assumption than in the deterministic one! This is differentfrom what we can observe for the blocking probability for the classical Erlang’s loss model; cfe.g. [101, Figure 8]. Moreover, the deterministic arrivals increase the number of outage incidents


Figure 4.5: Number of outage incidents as in Figure 4.4 for traffic 600 Erlang/km2.

for intermediate values of the SINR and decrease for extreme ones, especially with the fair policy.Concluding these observations one can say however, that the differences between Poisson anddeterministic are not very significant and hence the Poisson model can be used to approximatea more realistic traffic model.


Figure 4.6: Deep outage versus outage time. For any policy LESF(δ), with 0 < δ <∞, the leftcurve of a given style represents the fraction of time spent in deep outage. The right curve ofa given style recalls the fraction of time spent in outage (already plotted on Figure 4.2). Theoptimal policy (δ = 0) does not offer any “best effort” service. The fair policy (δ = ∞) offersthis service for all users in outage.


Figure 4.7: Mean total throughput normalized to its maximal value 256kbit/s obtained duringthe service time (upper curves) and its fraction obtained when a user is in outage (lower curves)for different policies LESF(δ) traffic 900 Erlang/km2.


Figure 4.8: Impact of the deterministic arrival process (as compared to the Poisson one) on themean fraction of the requested streaming time in outage, for the optimal and fair policy; traffic900 Erlang/km2.


Figure 4.9: Impact of the deterministic arrival process (as compared to the Poisson one) on themean fraction of the requested streaming time in outage, for the optimal and fair policy; traffic600 Erlang/km2.


Figure 4.10: Impact of the deterministic arrival process (as compared to the Poisson one) on themean number of outage incidents for the optimal and fair policy; traffic 900 Erlang/km2.


Figure 4.11: Impact of the deterministic arrival process (as compared to the Poisson one) on themean number of outage incidents for the optimal and fair policy; traffic 600 Erlang/km2.

Appendix

4.A A general real-time streaming (RTS) model

In this section we will present a general stochastic model for real-time streaming. An instantiationof this model was used in the main body of this Chapter to evaluate the real-time streaming inwireless cellular networks. This model comprises a Markovian, multi-class process of call arrivalsand their independent, arbitrarily distributed streaming times. These calls are served by a serverwhose service capacity is limited. Depending on the numbers of calls of different classes present inthe system, the server may not be able to serve some classes of users. If such a congestion occurs,these classes are temporarily denied the service, until the next call arrival or departure, when thesituation is reevaluated. These service denial periods, called outage periods, do not alter the callsojourn times in the system. Our model allows for a very general service (outage) policy sayingwhich classes of users are temporarily denied the service due to insufficient service capacity. Wewill evaluate key characteristics of this model using the formalism of point processes and theirPalm theory, often used in the modern approach to stochastic networking [11]. Specifically, weare interested in the intensity of outage incidents, the mean inter-outage times and the outagedurations of a given class, seen from the server perspective, as well as the probability of outage atthe arrival epoch, mean total time in outage and mean number of outage incidents experiencedby a typical user of a given class. The expressions developed for these characteristics involveonly stationary probabilities of the (free) traffic demand process, which in our case is a vector ofindependent Poisson random variables. Recall that such a representation is possible e.g. for thewell known Erlang-B formula, giving the blocking probability in the classical (possibly multi-class) Erlang’s loss model. Indeed, our model can be seen as an extension of the classical lossmodel, where the losses (i.e., service denials) are not definitive for a given call, but only temporal— having the form of outage periods.

4.1.1 Traffic demand

Consider J ≥ 1 classes of users identified with calls. We assume that users of class k ∈ 1, . . . , Jarrive in time according to a Poisson process Nk = T kn : n 9 with intensity λk > 0 and stay inthe system for independent requested streaming times W k

n having some general distribution withmean 1/µk <∞. All the results presented in what follows do not depend on the particular choiceof the streaming time distributions — the property called in the queueing-theoretic context in-sensitivity property. Denote by Nk = (T kn ,W k

n ) : n the process of arrival epochs and streamingtimes (call durations) of users of class k. We assume that Nk are independent across k = 1, . . . , J .Denote by Xk(t) =

∑n 1[Tkn ,T

kn+Wk

n )(t) the number of users of class k present in the system at

9The time instants Tkn are used only in the Appendix and should not be confused with Tk denoting in the

main stream of (and in the proof of Proposition 12 at the end of the Appendix) the mean throughput of a userin class k.

121


time t and let X(t) = (X1(t), . . . , XJ(t)); we call it the (vector of) user configuration at time t.The stationary distribution π of X(t) coincides with the distribution of a vector of independentPoisson random variables (X1, . . . , XJ) with means E[Xk] := ρk = λk/µk, k = 1, 2, . . . , J . Wecall ρk the traffic demand of class k.

We adopt the usual convention for the numbering of the arrival epochs T k0 ≤ 0 < T k1 . Thesame convention is used with respect to all point processes denoting some time epochs.

4.1.2 Resource constraints and outage policy

For class k = 1, . . . , J , let a subset of user configurations Fk ⊂ NJ be given, where N = 0, 1, . . .,such that all Xk users of class k present in the configuration X = (X1 . . . , Xk, . . . , XJ) are servedif and only if X ∈ Fk and no user of class k is served (we say it is in outage) if X 6∈ Fk. Wecall Fk the k th class (service) feasibility set. Denote by πk = π(Fk) the probability that thestationary configuration of users is in k th class feasibility set.

We assume that, upon each arrival or departure of a user, the system updates its decisionand, for any class k, it assigns the service to all users of class k if the updated configuration ofusers is in Fk. All users of any class j for which the updated configuration is in F ′k = NJ \ Fkwill be placed in outage (at least) until the next user arrival or departure.

In what follows we will assume that no user departure can cause outage of any class of usersi.e., switch a given configuration from Fk to F ′k. (However a user departure may make someclass j switch from F ′j to Fj .)

Denote by Xk(t) := Xk(t)1Fk(X(t)) the number of users of class k not in outage at time t.Denote by X(t) = (Xi(t), . . . , XJ(t)) the configuration of users not in outage at time t.

4.1.3 Performance metrics

In what follows we will be interested in the following characteristics of the model.

Virtual system metrics

During its time evolution, the user configuration X(t) alternates visits in the feasibility set Fkand its complement F ′k, for each class k = 1, . . . , J . We are interested in the expected visitdurations in these sets as well as the intensities (frequencies) of the alternations. More formally,for each given k = 1, . . . , J , we define the point process Bk := τkn : n of exit epochs of X(t)from Fk; i.e., all epochs t such that (X(t−),X(t)) ∈ Fk×F ′k (with the convention τk0 ≤ 0 < τk1 ).These are epochs when all users of class k present in the system (if any) have their serviceinterrupted.

Denote by σ′kn := supt − τkn : X(s) ∈ F ′k ∀s ∈ [τkn , t) the duration of the n th visit of theprocess X(t) in F ′k and by σkn := τkn+1 − τkn − σ′kn the duration of the n th visit of the processX(t) in Fk. We define for each class k = 1, . . . , J :

• The intensity of outage incidents of class k, i.e., the mean number of outage incidents ofthis class per unit of time

Λk := limT→∞

1

T

∑n

1[0,T )(τkn) .

Obviously Λk is also the intensity of entrance to the k th class feasibility set Fk.

4.A. A GENERAL REAL-TIME STREAMING (RTS) MODEL 123

• The mean service time between two outage incidents of class k

σk := limN→∞

1

N

N∑n=1

σkn .

• The mean outage duration of class k

σ′k := limN→∞

1

N

N∑n=1

σ′kn .

Note that the above metrics characterize a “virtual” quality of the service, since some visitsin Fk and F ′k may occur when there is no k th class user in the system (in the latter case theoutage of this class is not experienced by any user).

User metrics

We adopt now a user point of view on the system. We define for each class k = 1, . . . , J :

• The probability of outage at the arrival epoch for user of class k

Pk = limN→∞

1

N

N∑n=1

1F ′k(X(T kn )) .

• The mean total time in outage of user of class k

Dk = limN→∞

1

N

N∑n=1

∫[Tkn ,T

kn+Wk

n )

1F ′k(X(t)) dt .

• The mean number of outage incidents experienced by user of class k after its arrival

Mk = limN→∞

1

N

N∑n=1

∑m

1(Tkn ,Tkn+Wk

n )(τkm) .

Note that possible outage experienced at the arrival of a given user is not counted in Mk.The mean total number of outage incidents (including possibly at the arrival epoch) ishence Pk +Mk.

4.1.4 Mathematical results

For a given class k = 1, . . . , J , denote by εk = (0, . . . , 1, . . . , 0) ∈ NJ the unit vector having itsk th component equal to 1. Hence x+εk represents adding one user of class k to the configurationof users x ∈ NJ . Denote by Pr the probability under which X(t) : t is stationary and by Ethe corresponding expectation. Recall that πx ∈ · = PrX(t) ∈ · is the distribution of thestationary configuration of users X(t) (it corresponds to independent Poisson variables of meanρk).


General results

We present first results regarding the virtual system metrics. These results will be next used toevaluate the user metrics.

Lemma 8 The intensity of outage incidents of class k is Pr-almost surely equal to

Λk =

J∑j=1

λjπ x ∈ Fk,x + εj ∈ F ′k k = 1, . . . , J.

Proof. Let N =∑Jj=1Nj be the point process counting the arrival times of users of all

classes. By independence, N is the Poisson point process of intensity λ =∑Jj=1 λj . Then, by

the ergodicity of the process X(t) and the fact that the exits from Fk can take place only atsome user arrival epoch we have by the Campbell’s formula 10 [11, Equation (1.2.19)],

Λk = E

[∫[0,1)

1Fk×F ′k (X (t−) ,X (t))N (dt)

]

= λ0

PrNX(0−) ∈ Fk,X(0) ∈ F ′k ,

where Pr0N designates the Palm probability associated to N (which is, roughly speaking, the

conditional probability given an arrival at time 0). By the PASTA (Poisson Arrivals See TimeAverages) property [11, Equation (3.3.4)] the configuration of users X(0−) under Pr0

N has dis-tribution π. Moreover, X(0) = X(0−) + εξ where ξ ∈ 1, . . . , J is under Pr0

N independent ofX(0−) and takes value j with probability λj/λ. This completes the proof.

Lemma 9 The mean service time between two outage incidents and the mean outage durationof class k are Pr-almost surely equal to, respectively,

σk =π (Fk)

Λk, σ′k :=

π(F ′k)

Λkk = 1, . . . , J,

where Λk is given in Lemma 8.

Proof. First we prove the expression for σk. By ergodicity σk = E0Bk

[σk0]

Pr-almost surely,where E0

Bkdesignates the expectation with respect to the Palm probability associated to Bk,

and E0Bk

[τk0]

= 1/Λk; [11, see e.g. Equation (1.6.8) and Equation (1.2.27)]. Applying the mean

value formula [11, Equation (1.3.2)]11 we get π(Fk) = ΛkE0Bk

[σk0], which completes the proof

of the expression for σk. For the other expression, note by the definition of the sequence σkn, σ′kn

and τkn that Pr-almost surely,

σ′k = E0Bk

[σ′k0]

= E0Bk

[τk1 − σk0

]=

1

Λk− π(Fk)

Λk=π(F ′k)

Λk,

which completes the proof.

Proposition 13 The probability of outage at the arrival epoch for a user of class k is equal to

Pk = π x + εk ∈ F ′k k = 1, . . . , J (4.16)

Pr-almost surely.

10With Zn := (X(Tn−),X(Tn)) and f(t, z) = 1[0,1)(t)1Fk×F′k(z)

11with Zk (t) = 1Fk (X (t))

4.A. A GENERAL REAL-TIME STREAMING (RTS) MODEL 125

Proof. By ergodicity we have Pk = Pr0NkX(0) ∈ F ′k, where Pr0

Nkdesignates the Palm

probability associated to Nk (arrival process of the users of class k). By the PASTA property theconfiguration of users X(0−), just before arrival of the user of class k at time 0, has distributionπ. Once the user enters the system, the user configuration becomes X(0−) + εk, whence theresult.

Proposition 14 The mean total time in outage of a user of class k is Pr-almost surely equal to

Dk =1

µkπ x + εk ∈ F ′k k = 1, . . . , J . (4.17)

Proof. Again using the ergodicity of X (t) we can write

Dk = E0Nk

[∫[0,Wk

0 )

1F ′k(X(t)) dt

].

Denote by Y(t) := X(t) − εk1[Tk0 ,Tk0 +Wk

0 )(t) the process of configurations of users other than

the user number 0 of class k (which arrives at time 0 under E0Nk

). By Slivnyak theorem [9, see

e.g. Theorem 1.13] the distribution of the process Y(t) : t under Pr0Nk

is the same as this

of X(t) : t under Pr. Using the fact that W k0 and Y(t) are independent under Pr0

Nkwith

E0Nk

[W k0 ] = 1/µk we obtain

Dk =

∫ ∞0

E0Nk

[1[0,Wk

0 )(t)1F ′k(Y(t) + εk)]

dt

=1

µkπ x + εk ∈ F ′k)] ,

which completes the proof.

Proposition 15 The mean number of outage incidents experienced by user of class k after itsarrival is Pr-almost surely equal to

Mk =1

µk

J∑j=1

λjπ x + εk ∈ Fk,x + εk + εj ∈ F ′k , (4.18)

k = 1, . . . , J .

Proof. Again using the ergodicity of X (t) we know that, Pr-almost surely,

Mk = E0Nk

[∫(0,Wk

0 )

Bk(dt)

].

Using the fact that W k0 and Y(t) are independent under Pr0

Nkwith E0

Nk[W k

0 ] = 1/µk we obtain

Mk = E0Nk

[B∗k(0,W k

0 )]

=Λ∗kµk

,

where B∗k =: τ∗kn : n is the point process of exit epochs of X(t) from F∗k = x : x + εk ∈ Fkand Λ∗k its intensity. Using Lemma 8 with Fk replaced by F∗k concludes the proof.

We will now prove the result regarding the throughput of the typical call of class k.


Proof of Proposition 12. We have

Υk = Υδk = µkE

0Nk

[∫[0,Wk

0 )

rk1(X(t) ∈ Fδk)

+ r′δk (X(t))1(X(t) 6∈ Fδk) dt

].

It is easy to see, as in the proof of Proposition 14, that Υk = rkπx + εk ∈ Fδk

+ Υ′k, where

Υ′k = E[r′δk (X(t) + εk)1((X(t) + εk) 6∈ Fδk)

]is the part of the throughput obtained by a user of class k during its outage time.

Chapter 5

Conclusion and future work

This thesis aimed at developing the methods for the examination of the QoS perceived by usersin cellular networks. QoS for services such as data transmission and real-time streaming areexamined. The approach consists in decomposing the problem into three levels corresponding tothree time scales. Firstly, the single link capacity is studied on the ground of information theory,after that users’ arrivals and departures are considered using queuing theory. Finally, the spatialpatterns of network resources are taken into account using stochastic geometry.

More specifically, in Chapter 2 we describe a simple model of a MIMO cellular network whichpermits to obtain an analytical lower bound for user bit-rates which are feasible from the infor-mation theory point of view. This expression accounts for the variety of MIMO configurations(numbers of transmitting and receiving antennas) and radio conditions (SINR). We validate theanalytical lower bound by comparison to the results of 3GPP simulations and to measurementsin the field.

In Chapter 3 we establish the dependence relation between the traffic demand and mean userthroughput for large wireless cellular networks serving variable bit-rate calls.

Further, we evaluate the user QoS (spatial CDFs of mean user throughput per cell, meannumber of users per cell and cell loads. We develop two approaches: the typical cell approachcorresponds to spatial averages of the characteristics of the cells in a large network. Sincethe averages of some crucial characteristics do not have explicit expressions, we propose thealternative mean cell approach. It permits an explicit expression of the major characteristicsand approximates well the typical cell. Also a heterogeneous cellular network model allowingfor different BS types (having different transmission powers) is proposed, aiming to help inperformance evaluation and dimensioning of real (large, irregular) operational networks. Itallows one to identify key laws relating the performance of the different base station types.Wevalidate the proposed approach by comparing its results to real field measurements.

The dimensioning for streaming traffic as well as mixing such traffic with variable bit-ratecalls are important axes for future work. These studies raised also open theoretical questionsregarding the stability of spatially and, more difficult, space-time dependent processor sharingqueues modeling the performance of individual network cells (cf Section 3.5.3). More work isalso required to understand the problem of different performance of the network during day andnight hours (cf Figure 3.15).

In Chapter 4, a real-time streaming (RTS) traffic, as e.g. mobile TV, is analyzed in thecontext of wireless cellular networks. An adequate stochastic model is proposed to evaluate userperformance metrics, such as frequency and number of interruptions during RTS calls as functionof user radio conditions. Despite some fundamental similarities to the classical Erlang loss model,

127

128 CHAPTER 5. CONCLUSION AND FUTURE WORK

a new model was required for this type of service, where the service denials are not definitivefor a given call, but only temporal – having the form of, hopefully short, interruptions (outage)periods. Our model allows one to take into account realistic implementations of the RTS service,e.g. in the LTE networks. In this latter context, several numerical demonstrations are given,presenting the quality of service metrics as function of user radio conditions.

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